As others have pointed out several times already, It seems people may be thinking that they are sending iMessages and either a) it's really a regular text and they don't know the difference, or b) the iMessage service is failing somehow and sending as an SMS instead.
I personally have noticed an error where my messages were sent both as iMessages and as regular SMS texts. The iMessages didn't count against my txt limit, but the actual texts did. Yes, I was sending and receiving duplicates--and had I already gone over my texting plan limit, I would have been charged for those duplicate SMS texts. I suspect something similar is happening to OP.
This was obviously some error in the iMessage service. It is notorious for weird problems such as this. Until Apple fixes it on their end, there's not much any of us can do.
The moral of the story is that Apple is not very experienced with the cloud and online services, and until they hire or otherwise buy that experience and totally re-vamp their back-end, we're going to see weird ***** like this and we'll just have to live with it.
With the recent management shake-up, I'm very optimistic about this stuff getting sorted out in the near future.