Can I use a one time pad key twice with random plaintext? I understand the basics of OTP: $|\text{key space}| = |\text{plaintext space}|$ implies perfect security, key reuse destroys this.

Cryptanalysis on the $N$-Time Pad for $N &gt; 1$ involves finding patterns in the ciphertext; this, however, all seems based on the premise that patterns exist in the plaintext (e.g. English language).

My question is: if $m_1$ and $m_2$ are two truly random strings, and they are both encrypted with the same OTP key to yield $c_1$ and $c_2$, is it possible to recover $m_1$ and $m_2$ from $c_1$ and $c_2$?

In other words: while "perfect security" (aka perfect indistinguishability) is obviously lost (constructing a distinguisher for two messages is trivial), is the plaintext still safe?
Yes, encrypting two different random "plain texts" with the same "pad" is indistinguishable from using two different random one time pads for encrypting the same plain text. You get perfect secrecy in the latter case, so you will get corresponding secrecy in the former case as well.

However, usually there is a functional difference between the key and the plain text that the above argument doesn't account for. More precisely, the recipient will usually "do things" with the plain text once decrypted, and in that respect the above argument might fail. 

For instance, if the random messages are symmetric encryption keys for a cipher that is vulnerable to a related key attack, it is obviously not safe to transport those messages encrypted with a reused pad.