Is there a good plugin to restrict content and allow users to connect with their Facebook or Google info?
Basically you are going to create a membership site. In this condition either you will have to code a plugin or just install
- Membership Plugin (To hide content from non logged in users). You might not need this if you want to completely hide the post from public. Check Content Visibility - Private Post
- Social Login plugin (for logins, signups and getting their identity)
Both can be find easily in wordpress plugin dir. Depending on your needs you may choose any of them. There are also plenty of paid membership plugins, with a lot of features.
What content will Facebook and Google allow us to see/store on our end?
Most of the WP Plugins are designed for just social login and getting public profile photo, name and date of birth. If you want more then you might need to fork a plugin.
Facebook says: "Ask only for the permissions that you need". & for Google you may ask for public accessible data as well as some private data. But Yes, be aware, people may raise his eyebrow if you are looking for a lot of personal information.