They are both local (this is important, albeit not very helpful 🙂 ).
HEAD is your current branch, which means the one you have checked out now. For git merge that’s the one you had checked out when you started.
MERGE_HEAD is the other commit, which means the hash ID of the commit you told Git to merge. That is, git merge origin/master resolves origin/master to some local commit hash ID, and then merges that commit, and MERGE_HEAD contains the hash ID of that commit.
I think a better term for the other commit is other or --theirs, and Git sometimes uses those terms, but other bits of Git do use the term remote to refer to the --theirs commit.