It depends, for most local apps you just have to log-in once while in upgrade mode (read-write on base system and overlayfs disabled), then the cookies will persist indefinitely and you'll be logged-in whenever you boot your computer in read-only mode.

For websites the cookies tend to expire once every few months so I'll boot back to upgrade mode and refresh my log-ins everywhere at once every few months at the same time as I install my system updates.

You can also selectively copy some software's local data to your persistent read-write disk partition and create a symbolic link in the original directory, that way this particular piece of software will be white-listed in a sense and always writable. But I haven't needed to do that.

My only issue has been google which has a weird cookie implementation, it throws cookie-mismatch errors all the time so I have to delete them and relog-in every day. If it was a big issue I'd make my firefox profile always writable as described above but I don't use google much.