It is said that the Deep Space Network can detect low-power signals sent from Voyager in the range of 10^-16 W. I can't seem to find any sources that indicate what the minimum power requirement of a received signal is. Also, what would be the minimum required power for the DSN to reliably interpret the signal? Does this figure differ between the 70m and 34m dishes?
There's no minimum power for any receiver if you don't specify the type of transmission and the amount of knowledge about that existing at the receiver!
For example, a GPS receiver has way worse characteristics (for example, a 4-bit ADC) than a digital TV receiver. So, who needs higher power to work?
GPS works well below the noise floor because the data rate received is so low, and the receiver knows what to look for very exactly (mathematically: it uses correlation over a long spreading sequence, meaning that it does an inner product in the number-of-samples-dimensional complex (signal) vector space, which means that the magnitude you get when detecting signal-containing noise is far, far higher than when just doing the same product with a vector that's purely white noise).
TV reception, on the other hand, needs to be high-rate, and thus, the SNR must be way, way better (see: Shannon Capacity C = 1/2 B · log_2(1+SNR) ).
Since the DSN isn't tailored to one specific type of signal but equipped with hardware and software that can be configured/programmed to pick out a desired wavefrom, the sensitivity depends on the transmission type and can't be generally answered.
The only figure of merit you could get would be the system's Noise Figure or the equivalent Noise Temperature, because that's how much worse a signal's SNR gets by going through the system. That would allow you to calculate the theoretical sensitivity for any given transmission. Now, since DSN is extremely distributed, especially among hot and cold objects, that cumulative number can't exist either, sorry.