It seems like you have a local implementation on your local machine, and you push your code to github. You also have another developer who have access to the github repository and will pull your changes on his/her machine.
if both of you run Strapi on your local machines, that means you have separate databases. So, maybe both of you should add the path to the database file (config/database.js) to the (.gitignore) file so you don’t disrupt each other.
If you have a shared database, which you can do if you set up a database in AWS RDS (for instance), and you both use the same database credentials regardless on where you install Strapi. I assume you know that you can separate the Strapi server from the DB. Hence, this option is valid.
Regarding your question how where users would login, that would be something that you need to develop on your own. Remember, Strapi gives you 2 groups of users:
Strapi backend users: here you have limited number of roles and users. This kind of user is someone who can perform actions on the Strapi admin panel. This kind of user should not perform actions on the frontend as any regular user.
Strapi frontend users: here you can get access to the APIs and use them. you can specify what API should be accessible by which role. This kind of users will not have access to the Strapi Admin panel. You will need to develop your own frontend login page to use the login/registration API.
If you’re starting new in Strapi, I recommend that you invest an hour to go through the documentation of version 4. Also, there are a few nice videos on Youtube if you search for (strapi v4 crash course).