If you're impressed by the work that somebody's shared and want to be notified when they upload new scores, you can click the "Follow" button that appears towards the top right of their profile and on any of their scores. Similarly, anybody can follow you.
When you're following someone, you'll be notified on your dashboard when they post a new score—and when they follow somebody new (with an option for you to follow the same person).
Who follows who is public information. You can see the number of people who follow a given user and the number of people that user follows on the right of their profile, and by clicking the "Followers" or "Following" number you can see the whole list.
You can see your own followers and people you follow in a similar fashion, by looking on the right of your profile (https://musescore.com/user), or by going directly to https://musescore.com/user/followers and https://musescore.com/user/following.
If you no longer want to follow somebody you currently follow, click the "Unfollow" button that appears in the same place the "Follow" button originally did.