They didn't have two functioning control rooms, so swapped off using the control room for the vehicle that was most active at the time, and used the distributed Mercury-style control for the less active vehicle, swapping back as required.
The men in John Hodge's Flight Control Division found it "a hell of a
great challenge and to a man they wanted to press on as soon as
possible." One of them suddenly said, "Why don't we handle it as if
one of the spacecraft were a Mercury-type and the other a Gemini-type
spacecraft?" Mercury controllers at the tracking stations observed
data on their consoles, summarized it, and forwarded the result by
teletype to Mercury Control Center. Gemini VII could be handled that
way while it served as a passive target for Gemini VI. For Gemini
missions, the stations were fitted with computer communications
processors. As the spacecraft passed overhead, the processors
interrogated the appropriate systems for specific data, which were
automatically transmitted to Mission Control. Gemini VI, the
active partner in the rendezvous, would be controlled by the more
sophisticated system. With this as a basis, an operational mode was
laid out.
After Gemini VII lifted off, flight control would be carried out in
the normal manner while the pad was being prepared for the second
launch. Once the flight controllers were sure the orbiting spacecraft
was operating properly, Mission Control would concentrate on Schirra
and Stafford in their spacecraft, and the tracking network would watch
Gemini VII, record data, and send information by teletype to the
Houston controllers. This mode would continue until the complicated
rendezvous mission ended and Gemini VI-A (so called to distinguish it
from the originally planned mission whose objective had been
rendezvous with Agena) returned to Earth. Then Gemini VII would become
the focus of communications again. Kraft was soon convinced that the
operation could be carried out safely. He told his Mission Planning
and Analysis Division to set up the flight plan so the second launch
could take place as soon as the pad was ready.
source
Some details about the means used to configure the control center are described on pp. 184-185 of the Gemini Midprogram Conference