There is a set of standartized web protocols called Webfinger, ActivityStreams, Salmon and PubSubHubbub which allow websites to interact with each other as a sort of decentralized/federated Twitter (this is called the "fediverse").
Disclaimer: I'm working on Mastodon, it's a new open source implementation, I'm trying to make it more approachable to people who value UX and maintainability
Short answer: The sidebar has a form at the bottom where you can enter a username@domain to follow them. For that method, you have to know the user in advance, realistically atm you can browse the public timelines on quitter.se and quitter.no etc to see who interests you (or ask friends who're using the network, etc).
I'm going to eventually implement a "who to follow" suggestions section to make this process smoother.
Mentioning people in statuses works the same as in Twitter, except if it's a user on the same domain as you (e.g. mastodon.social) you can enter their @username, but if they are somewhere else, you enter @username@domain.
If you are replying to a status on your timeline, the mentions are pre-filled like you would expect from e.g. TweetDeck to save you the typing.
Edit: You can follow me as Gargron@mastodon.social
Parent post's product, Sublevel, would make for a nice public communication utility.
Unfortunately, while I like it, it's definitely not going in the same direction as Twitter and is not likely to match everybody's idea of a social network. While its current members, myself included, like it for its simplicity and lack of embellishment, it is precisely because it comes with fewer features and deliberate design choices against janky embedded media, it does not and likely will not represent a Twitter alternative for the vast majority of Twitter users.
Sublevel could however be made into an interesting, social-first alternative to Pinboard.
If Sublevel were for sale, that's likely what I'd do with it, offering an archive utility to save HTML where duplicate content is only saved once and blindly shared among users to conserve space. Then stick an incrementally increasing price tag on membership like Pinboard. Maybe membership tiers.
I remember as well numerous decentralised alternatives to Twitter like Diaspora and Twister IIRC.
Twitter is horrible to use and would benefit from a complete rethink in terms of UI, maybe something along the lines of Opera Coast.
There is a set of standartized web protocols called Webfinger, ActivityStreams, Salmon and PubSubHubbub which allow websites to interact with each other as a sort of decentralized/federated Twitter (this is called the "fediverse").
[1]: https://gnu.io/social [2]: https://github.com/Gargron/mastodon
Disclaimer: I'm working on Mastodon, it's a new open source implementation, I'm trying to make it more approachable to people who value UX and maintainability