Thanks Taylor!
Not at all sure as it turned out, even more, that definitely wasn’t something I wanted to use. Oh well, my fault really, done it all too fast, only skimmed through the documentation instead of reading it properly, and even the name of “streaming” API didn’t ring a bell.
Anyway, even before realizing this I have switched to statuses/user_timeline and magically everything started to work, so I’m all fine now.
Thanks again!