It's insecure precisely due to the reason you mention - there is some information leakage.
Basically, if you have any assumptions about plaintexts (english text, files with known structure, etc), it leads to an easy statistical analysis. Probably using it twice doesn't change the practicality of the attack significantly, but using it many times with a non-random plaintext, eventually reveals enough information to recover the key.
Finally, if you have the ability to use it only twice, you also have the ability to use it only once - the restriction is that these one-time-pads are not to be used potentially unknown and over time, damaging number of times.
At one extreme, if you're aware (known plain-text) that the $m_1$ is just a null-string of the length of the pad, you have handed the attacker the key prior to computing $m_2$.
Known plain-text attacks are fairly common, it's reasonably easy to coerce an encryption mechanism to encrypt something you know a-priori. If not, you can usually make reasonable statistical assumptions.