This is such a dumb question. There are many space organizations. Let's take ESA for this case. Given the two Voyager spacecraft are transmitting a weak radio signal to the Earth, can ESA command and receive those signals and later calculate the location of the spacecraft and get data of thermoelectric generator conditions, others, etc. independently without the help of NASA, or is there some form of signal encryption that would require ESA to undertake certain action? Provided that ESA can use the Deep Space Network?
There isn't any encryption. As noted by this CCSDS report, "In the past, civil space missions relied on their uniqueness to deter unauthorized access" rather than encryption. Also, the diagrams in the following descriptions leave no space for an encryption function: Voyager comms and Voyager computer.
Error correcting codes are used, but these are well known (see this answer). The main problem would be knowing the format of the telecommands and telemetry packets. Decoding the telemetry is relatively easy (see Decoding Voyager 1) - knowing what the decoded message means without some information from NASA would be tougher. Similar for the telecommands.
I'm pretty sure that NASA/NASA JPL who created the Voyagers are the mission control for them. Just as ESA only controls spacecraft made and maintained by ESA.
The DSN is technically NASA's and they let other organizations use it because they don't have their own personal DSN. So just being able to use the DSN doesn't mean they can send data to any spacecraft that uses it.