On iPhone, you can turn off “Personalized Adverts”. You still get adverts but they’re not based on a profile of you.
Check the Google Store page of the app and read the Permissions/Privacy section. An app can’t do much without requesting relevant permissions first.
Apps by default have access to almost nothing. I’m an Android developer and know almost nothing about my users, because my app didn’t ask for those permissions.
I think it starts pinging google servers every few minutes in the background. Aside from that, no issues I can think of.