Ask HN: What are the realistic privacy concerns of using a PaaS

I've been looking at some PaaS offerings for my next app, and as it's main value is in the data it collects, I'm considering keeping the data on my own infrastructure.

So a quick question to those with more experience than me: is using something like Firebase going to hurt me down the line? I'm paranoid that if this thing goes well, Google would basically already have my data + a lot more resources and could simply leap frog wherever I'm at. Is this a genuine concern?