I do this no problem, it may not be audiophile quality but when you are streaming it doesn't matter so much because of the limited bitrate, I'd not do this if I was recording by the way. I have a dynamic mic with the gain at about mid way and the speakers behind the mic so they are not in the main path where the mic picks up audio. I have a noise gate as well and it works OK as far as I can tell. You do have to have the speakers at a low volume which is fine for the kind of game I play but probably wouldn't be loud enough if you are playing online FPS and need to hear footsteps etc. You can check out the quality on my most recent streams and see if it's OK.
Of course as with all these things it's about balance (excuse the audio pun) Everyone above is correct it totally depends on what you want to achieve, if you want crystal clear mic audio then don't use speakers, if you want to try it and see if it works the three key things I found to tweak were mic type, mic position (near your face and as far away from the speakers as possible) and speaker volume.