The personal information security concern I had is that it seems that pushbullet shovels all sorts of data from Chrome to the pushbullet server and then routes it to, pushbullet says, my other devices while respecting my privacy. While I don't doubt that pushbullet is an honest broker of my data, it's doing all this outside of google's purview. For somebody to spy on my data, they wouldn't need to break into google's ecosystem, they'd just need to break into pushbullet's.
I'm not disagreeing with all the other comments here about big bad google, I already think they are bigger and badder than everybody else here does.
And I'm not at all sure that what I'm pointing out has anything to do with google's motivation here (I liked the comment that said that this app is a threat to their walled garden), I'm just pointing out my impression to try to be helpful to OP in figuring this out.