There are two alternatives to Twitter and their ilk:
The federated approach, where many Twitter like sites can interconnect and exchange messages. One current example of this is Mastodon, other is GNU social (status.net/identi.ca).
The indie approach, where everyone basically hosts their own profile and there are no Twitter like sites at all, one instead uses readers that are completely separate from ones hosted profile, just like it was in the blogosphere days. One example of this is indieweb.org.
I tend to favor the indie approach as the federated one seems to in practice often end up with interoperability issues and mono-cultures, at least historically, plus there are still central providers whereas in the indie world everyone is a provider themselves.