"It I have a profile picture and I think it's obvious from the way I look that English is not my mother tongue"

That might have been true a century or two ago, but today's world is very multicultural. You could easily be Australian, American, British etc. I was actually under the impression you were a native English speaker until you said you weren't!

You said you would exchange French or Italian for English. Could those people have been native French or Italian speakers who were just practising their English with you? 

BTW, and not to argue with you (I assume that we are dicussing people expectations in general, and not the behaviour of those people):

Of course, proportion of English learners in the world and the same with regards to 'the Internet users' are two different fugures. I was exposed to English first in the form of video-games and programming languages... And question of ones ability to enjoy internet without getting some English aside (there are prominent  'diasporas' from some countries even in the Russian major social network) - is it easy to get to that 'another penpals website" without at least rudimentary English?

You open your Google.ru (which in addition to being tuned for Russia had 'show results in Russian' as the default settings) and type desired combination of words. First in Russian, then in your target lanaguge. They show you... italki, though its position isn't that close to top. Then you go to google.com. and it redirects you to google.ru. Then, you manage to return to .com. but now it still shows you results in Russian. You go to 'settings' and voilà.
Now you can type your line in English and see more or less the same as Americans.

But let's suppose, you do nothing of this (and what is most important - don't ask Google in English, the rest is for maniacs). Would it show you the desired penpal website?

***
There are a lot of Francophones and Italians who don't speak English around.
have you tried to look for "Échange linguistique", for example?
or "tandem linguistico"?
The first search brings immediately http://www.echangeslinguistiques.com/ and probably some other sites.
Search within .fr domain, specific search for Italian or French people interested in Indonesia.... There are lots of options:-/

My guess is that it's because English is considered THE international language, especially for international business. So most people assume that even if you are not a native English speaker, you probably understand and speak some English. I agree with Jmat, that nowadays countries are becoming more multi-cultural so people no longer have the immediate assumption that only caucasian/western-looking people speak English. I certainly never ever had that assumption as I myself am an asian-looking Singaporean Chinese and the most commonly used language in our country is English (it is taught as a first language in schools).

I can't speak for other countries, but I think in America it's very common for people to assume others can speak English. I think this is partly because America is very diverse; it's nearly impossible to tell if someone speaks English just by looking at them. But I think that's not the only problem-- Americans are also used to everything around them being in English, and many of them have not spent a long time studying a foreign language.

Jmat I was actually talking about the non-french and non-italian speakers who tried to start a conversation in English (on second paragraph). My question is why would they start to speak English to someone whose profile is not written in English and doesn't even look like an English native speaker. Is it because people just assume everybody speaks English these days?

However you said you were under the impression I was an English native speaker, so maybe that's one explanation. I wasn't just talking about me, though. I bet other people also have the same experience as I.