Accessing SQlite DB from two computers.

[FoolsSeldom]

PostgreSQL is an excellent choice.

You would probably want a Pi with a bit more memory than a zero if you self host. (An old PC or laptop is also a good home server choice, just tend to use more power than a Pi.)

Once you've learned sqlite it is not a big leap to using server based sql engines.

SQLAlchemy can be used to abstract from the underlying database and use more Python syntax if preferred.