Well, according to Merriam-Webster.com:
Bimonthly, adj:1. occurring every two months2. occurring twice a month
No wonder I'm confused when I hear something happens "bimonthly"! I guess you're supposed to tell the meaning by the context? The same goes for biweekly, biannually, etc.
As far as I can see, the word originally (1846) was supposed to mean "occurring every two months", hence the prefix "bi" (meaning "two, twice, double," etc.). I imagine that at some point people got confused when they heard "twice monthly" and started using it to mean twice per month, too. And you know how the English language works - enough people use a word (even if it's wrong) and that defines it.
In case you want to be "more" correct, you can try these alternatives for twice per month:
Fortnightly (since a "fortnight" is 14 days)