There’s Calibre for a local option. It doesn’t have an android app itself, but it will connect to android and manage your ebooks. https://calibre-ebook.com/

Calibre to manage your library, and Calibre content server to essentially make it available for other devices.

For web browsing and reading, one can use Calibre-web.

For Android, Librera reader is suitable, IIRC, on the Pro version, which you can download for free from Github, you can use Calibre as an OPDS and download books from your Calibre directly on the app.

Privacy versus convenience… For what I know, the most common solution to your concern would be to setup a Calibre server and access it via your Android devices. But that’s obviously a tad more complex than uploading EPUBs to Google Books.

• KOReader
• Librera Pro on F-droid

I assume you’re asking because you’re trying to get away from corporate/proprietary solutions, but in case you’re not: kindle has web reader and an Android app and lets you upload epubs sourced outside of the Kindle ecosystem.

BookFusion is cool. It also has a calibre plugin so you can upload directly to it if you use Calibre. No idea how they are privacy wise but the sync is perfect across multiple devices and systems. It is not open source but it does what you want and has paid levels. Here’s the link: https://www.bookfusion.com/

Sadly no dedicated app yet, but Kavita is a really great book server you can self host.
The webapp also works pretty well on Android until a full app can be deVeloped.

Despite the name, audiobookshelf recently also added e-book support. It is still in development but (at least for me) is at a point where I’ve abandoned kavita and am now using it for both my audio, and e-books.

You might be interested in this : https://readwise.io/read

I personally like though I mostly use it for RSS and as a read-it-later.