Using digital means, they are taking practical steps to offer desperate men, woman and children a place to stay in their own homes, or seeking to pressure their own governments into offering sanctuary to more of those in need.
And in Germany, a website has been running for months which aims to match offers of accommodation in private homes -- ideally shared rental apartments -- across the country with individual refugees in need of a place to stay.
Such direct action couldn't be more needed.
Migrants are pouring over Europe's borders in record numbers this year, according to the EU border agency Frontex
, many of them fleeing conflict in Syria, Iraq and Afghanistan. In July alone, a record 107,500 were detected at EU borders, it said.
'Refugees are our future spouses, best friends'
In Reykjavik, Bjorgvinsdottir's inspiration came from a friend who posted a status update on Facebook -- addressed to Iceland's Minister of Welfare Eyglo Hardar -- saying he wanted to take five Syrian refugees into his own home, she said.
With just that gesture, her friend pointed out, the number Iceland took in could increase to 55, Bjorgvinsdottir told CNN's "The World Right Now."
Struck by his idea, she thought that she could offer to pay for the flights. "Then I got this idea that maybe more people want to join in and give food or clothes or even offering their houses or extra bedrooms -- and I did that just to see how much we could raise the number from maybe 50 to 100 or 200," she said.
The idea took off and the page already has 12,000 members, Bjorgvinsdottir said -- no mean feat given the country's population is only about 300,000. Proportionately, that equates to some 12 million people signing up in the United States.
A post on her Facebook page says, "Refugees are human resources, experience and skills. Refugees are our future spouses, best friends, our next soul mate, the drummer in our children's band, our next colleague, Miss Iceland 2022, the carpenter who finally fixes our bathroom, the chef in the cafeteria, the fireman, the hacker and the television host. People who we'll never be able to say to: 'Your life is worth less than mine.'"
Bjorgvinsdottir hasn't yet been able to sort through the comments posted on the page to calculate how many offers of places to stay there are. But with the help of volunteers she hopes to pass that information on to the government in the near future, in the hope it can be put to concrete use.
The war in Syria has been dragging on for years, she said, and Icelanders don't understand why they are not seeing more refugees being given sanctuary in their country.
"People just really want to help and they really want to see actions -- and actions taken immediately, right now, because time matters a lot in this situation," she said.
Inspired by the Icelandic example, a U.S. group has also been set up on Facebook, called "Americans Supporting Syrian Refugees: Open Homes, Open Hearts."
Berlin host: 'Fantastic' experience to help a migrant
In Germany, Katie Griggs, a Briton who's been living in Berlin for several years, is one of those who responded to the call of Refugees Welcome and registered with the website in March.
She and her husband own their apartment, so couldn't provide an apartment share, but could offer a guestroom for emergency accommodation.
Just a few weeks after they signed up, they welcomed a pregnant woman from Nigeria into their compact, two-bedroom flat in Berlin's hip Kreuzberg district, Griggs told CNN by phone from Berlin.
The woman, whom Griggs refers to as Alyssa on her blog
to safeguard her privacy, had been referred to Refugees Welcome by another group in Berlin which helps trafficked women. Realizing she was pregnant and in a worsening situation, Alyssa had left Greece -- where she'd spent two years after traveling from Nigeria -- and made another perilous overland journey to Germany in search of help, Griggs said.
"Of course we had worries and then other people throw worries at you," said Griggs of her initial reaction on being asked to host Alyssa in May. Those concerns included questions over the penniless 24-year-old's health and their own personal security.
But, Griggs said, "The actual experience was brilliant, it was absolutely fantastic. We got to learn about Africa just in our own living room. She was lovely, she was so positive the whole time, even though she was on her own, a pregnant woman.
Alyssa told how she'd left school at 11 because there was no money to pay for her education, Griggs said. Griggs tried to help their guest with German and with learning about the baby she's expecting in November.
"But she also gave us lots back -- we learned about Nigerian food, about Nollywood, she would do a bit of dancing for us, teach us about African music and culture," said Griggs.
Six weeks later, Alyssa felt able to lodge a formal asylum claim and was whisked off by train to a center in western Germany the same day, clutching her meager belongings in a plastic bag. Since then, Alyssa has stayed in touch by phone -- and Griggs says she misses having her in their home.
Berliners lend a Syrian refugee a bike
Horrified by the images she's seen of conflict and the tragic deaths of Syrian refugees seeking to reach Europe, Griggs has also tried to help Syrian refugees in Berlin.
She signed up to a Berlin-based Facebook bicycling group
set up by a Syrian refugee, Monis Bukhari, to help fellow Syrians arriving in the city adjust to their new lives.
It asks Berliners to lend bikes for one-day bike tours, so that the newly arrived refugees can meet fellow residents of the city, German or other nationalities, stay active and get to know their way around -- all helping them to integrate into the community.
The cycling group brings together mostly people aged 25 and under, Bukhari said, with similar interests -- important when so many of the Syrians coming to Germany are young.
And it's not a one-way street.
Bukhari, who arrived in Berlin two years ago and has recently been granted a German passport, is using Facebook to organize a "Thank you Germany" event
on October 10, in which Syrians will hand out flowers randomly to Germans at stations across the country and play music for free for their enjoyment.
Bukhari, who's part of a wider volunteer group within Germany called Syrian House which seeks to help new arrivals navigate German laws and bureaucracy, told CNN that the event has a dual purpose.
"First of all, we as Syrians, we think that German people deserve to be thanked because they are supporting us and helping us, and they showed so much sympathy," he said. This is true even when the people involved share no common language, he said.
"The other side is that we in Syrian House think that this way of thanking will help or, let's say, push the people who refuse refugees to accept them, to accept the idea of welcoming refugees."
Bukhari, who ran his own design studio and publishing firm in Damascus, knows how vital that welcome can be. He first fled Syria for Beirut and then Jordan after he was accused by the Syrian authorities of being a spy and sentenced to death in absentia. He was invited to work in Germany on a project and has stayed there since after Jordan declined his return, he said.
Next month Syrian House will launch a photography exhibition and book by Syrian photographers documenting the refugee experience in Germany, and the kindness of some people they encounter, Bukhari said.
Another Facebook page advertizes the first concert by the Syrian Expat Philharmonic Orchestra
-- which describes itself as "the first symphony orchestra for the Syrian professional musicians who live in European Union countries" -- on September 22, in the German city of Bremen.
Merkel: Germany must show 'flexibility'
Amid the turmoil of recent months, as European nations -- particularly Greece and Italy -- struggle to cope with the migrants reaching their shores, Germany has stood out as one of the more welcoming EU members.
Germany's government said last month it expected up to 800,000 asylum seekers to come this year -- four times more than in 2014.
German Chancellor Angela Merkel said Monday that her country -- where some are opposed to taking in asylum seekers -- must show "flexibility" when it comes to dealing with the crisis.
For the most part, the public supports Merkel.
Local football clubs hoisted "welcome" banners over the weekend. Villages held "refugee welcome" parties for the newcomers. And a recent news poll estimated that 60% back Merkel's warm welcome.
But not everyone in Germany feels the same way. There have been xenophobic protests, and a planned asylum center was burned down last month.
Police in the town of Cottbus have arrested a suspect in an attack on about 40 asylum-seekers at a shelter in Brandenburg on Tuesday evening in which some kind of spray was used.
It's not clear what the substance was or what triggered the attack but several people, including children, were brought to the hospital for treatment. None are in serious condition, police said.