Given the breadth of ways to leak information about the private keys--side channel attacks, physical attacks, userspace (allocator, random number generator) this would be extremely difficult (impossible?) to prove.
When using an OTP, you have to use non-pseudorandom values to avoid just being a stream cipher. If you're doing that, you can skip sharing the pad and just share the initial state of the PRNG.
If you go to the trouble of sharing the pad, go to the trouble of using random data within it. :)