One certainly can! The MusicBrainz server software is available for download, licenced under the GPL v2. The data in the MusicBrainz Database is likewise available for download, some as Public Domain, some licenced CC-BY-NC-SA. And, one can subscribe to updates of the database, to keep your copy of the data in sync with the official MusicBrainz.
I do this myself, from time to time. Primarily my purpose is to explore the data in ways I can’t do through the public API. For instance, one time I got interested in how many leading characters of an MBID did one have to use to avoid ambiguity in the current set of MBIDs as of that moment.
There’s no automated way (that I know of) to send data upstream from a private copy to the official MusicBrainz. It may happen in special circumstances with the help of MB staff.
From time to time I wish for a way to store data that is similar to MusicBrainz, but not appropriate to contribute. For instance, I have some informally authored CDs that I listen to, but which were never publicly released. I would love to have a private MusicBrainz where I could store that data for my own use, which would delegate to the main MusicBrainz for all non-private content.