Our projector is close enough to the speaker that if he wants, he can connect his laptop to its second VGA in port. Some do that, others just agree a sign with the media team for when to change slides.
If you let the speaker control your media PC with Microsoft's Remote Desktop Connection, your media PC will no longer output anything useful on its video outputs - probably just a login screen. And you'll be locked out of the media PC until the speaker logs out...
Instead, you could use the Screenmonkey Remote Desktop plugin on the media PC, and a VNC server on the speaker's PC. That way the speaker is in control of what his own PC is showing, and you are in control of what the projector is showing - you can choose his screen, or then something else. You'll want to check the settings and connection beforehand, for any problems with lag, colour depth, animations etc.
Since you apparently have network connectivity between the two PCs, you could also look at the various kinds of software that allow just keyboard and mouse commands to be sent over the network, e.g.
MouseWithoutBorders. The presentation would run on the media PC, and only the input commands would travel over the network. The speaker would see the results of his actions on the projector (or you can set up a monitor that the speaker can see - we do that from the projector's VGA Out).