To extend PearsonArtPhotos answer, various amateur and academic groups successfully received Apollo voice traffic so it seems safe to assume the USSR would have been doing the same. Of more interest would have been the scientific and spacecraft health information coming back in telemetry, both for the science data and for feeding into the Soviet moon program engineering. The telemetry appears to have been unencrypted but certainly complex. Data extracted from it would have been of similar value to an unmanned lunar probe that you did not have to pay for, so spending serious effort to unpack it would be worth while, and would become easier as the post flight papers were published and Apollo/Soyuz flew.
In terms of radio based measurement the crafts all had transponders, including the boost stage which reduces needed power, but reading the wiki page it looks like only one station at a time could be interrogating so a third party popping in to get their own read would have jammed the NASA one so would have been both obvious and aggressive. Doppler measurements would have been possible to extract the craft velocity with respect to the ground station even if range was not, and post flight analysis would allow things like lunar orbit height and presence on lunar surface to be verified.
Using an active radar at the moon range is certainly possible now but the link suggests the signal processing techniques used are new. Doing radar ranging of the large departing first stage+lander+CSM near Earth would have been both possible and have provided hard numbers on the engine performance.
The Wikipedia page for unified S-band provides a broken link to a Russian site that may give more information if relocated.