Personally, I'm running BTSync - and even though I've got it syncing EncFS encrypted data, the app has enough privileges to read the unencrypted versions of those EncFS filesystems if it were instructed to.

I'd feel happier if a few trusted security experts from a few different countries/jurisdictions had blogged about their analysis of the source code and the likelyhood of the binary produced from the source being either intentionally or unintentionally compromised.

Having said that, as you point out, we've got the Chrome source, lots of people look very hard at it, and it _still_ fails year after year… Hopefully, BTSync and BTChat (or my hypothetical Open Source reimplemetations) are significantly less complex than a full featured browser, and not would not require nearly so much focus on performance that "provably secure" or perhaps just "significantly less likely to have obscure bugs" coding techniques could be used in spite of speed penalties - browser vendors have significant motivation to optimise for speed above all else, hopefully the much smaller subject domains of sync or chat clients would allow security to sensibly be prioritised instead.

A program might be open source, still the binaries offered for download might be compromised. Who is able to notice that now?

You might compile the software yourself, but the majority of users wont.

You might have reverse engineered a closed source software, but I guess you won't do that for the OSS binaries.