Hello Bob,
This link is to a web page that describes the functions of the pins in the 7 pin road lights plug and socket.
http://www.towingandtrailers.co.uk/shop/132/133/142/
Assuming pin 6 (red) becomes 12V positive with respect to pin 3 (white) -ve return when the foot bake is applied then both stop lights on the caravan should illuminate.
If only one comes on, it proves the cars wiring is OK, but it then suggests that either one of the bulbs on the caravan has blown or there is some fault with the carvans wiring. The Stop lights are usually wired in parallel, some manufactures simply put in a link wire between the light fittings in side the end wall of the caravan, others take it back to a terminal block.
Check the bulbs, either with your multimeter on continuity, or by swapping the bulbs over. If the falt switches sides, then its the bulb, but if the fault stays with same fitting, then its circuitry.
A very common problem with trailer electrics is corrosion in the light fittings and multipin plugs. It often worth pulling off and re-connecting any spade terminals and re-tightening any screw connections to break and clean corrosion away. The addition of a little Vasaline or grease to the spade before reconnecting can help to keep the contacts clean in the future. Its also worth looking at the bayonet sockets the bulb fits into, sometimes rust can set up just where the bulbs bayonet pins lock into the fitting and this can be just sufficient to break the electrical circuit and stop the bulb from working.
If none of that works, the it will be case of tracing your way back through the wiring to try and find any other possible faults.