This was easy to check--I opened a phone line connection with my home landline, then created, sent and received an SMS while on the phone. Using a Verizon Tour. So CDMA can perform this function.
If I remember correctly, SMS uses the control channel that the phone communicates with the network using. It is always live so you should be able to SMS and MMS during a phone call. Mine even works when I am tethered to my laptop.
I have little (or no) understanding of the new LTE technology that is coming to Verizon at least, but I believe at that time the newer devices will allow simultaneous data and voice. This I am not 100% on though.
On Sprint currently and Verizon last year, I have always been able to SMS while talking.
For me on my Curve, I could talk on the phone, and text for the first like 10 minutes or so, then the texts would not go thru anymore. I haven't tried it with the Tour yet. I would actually tell the person I was talking to on the Curve to call me back in one minute, so I could start the whole "10 minutes of texting while on the phone" thing again, lol.