Are you accessing it through a mapped drive letter, in place on the
other machine? Or via a UNC path? I don't see why a mapped drive
letter would matter, since that's always resolved to the UNC name,
but maybe it does cause a problem.

The issue I'm getting at is that if the replica thinks it's been
moved when you open it, then it will become a replica and no longer
be the design master.

Well, create a replica from the DM on your machine and in the
process transfer DM status to it. Of course, the other person won't
be able to make design changes to the replica on his machine any
longer.

I'm not exactly clear on what you're doing, though, so I may be
misdiagnosing the whole thing.

There should be no problems with storing a DM on a server and
working with it from any workstation connected to the server. That
also counts for when the "server" is actually just a workstation
sharing files.

The only thing I could think would be if you're not opening it in
place but actually moving it to your machine before trying to use
it. If you do that, yes, it will no longer be a DM, and you're to
blame for it losing DM status. The only way to move a DM legally is
by using the MoveReplica command in Replication Manager or in the
TSI Synchronizer. The other legal method is to transfer DM status
through a synchronization, but that's not a move, since it requires
two replicas already in place, and leaves the old DM in its original
location as a plain old replica.

I'm babbling in the hope that I'll say something that will strike a
chord and make sense to you so that it explains your problem. But it
may just be gibberish, after all. :)

Jet Replication has been depecrated for a decade

move to SQL Server replication