No because Sprint use a technology known as CDMA EV.DO for data while AT&T use GSM. You could only talk and surf the web using a Sprint Wimax phone and future LTE handsets.
I was wondering the same thing, so got on the Sprint chat and asked. I was told that yes, you can talk and access the the web due to the multitasking capabilities of the phone.
Obviously this is a poor design. I dont understand why this technology is so limited. Im sure programming can make it happen. I guess nobody should expect to use their new sprint Iphone for business. You cannot even check new email while talking. Or you cannot even check the web either. Plus where would you find wifi when driving. You may ask "who cares". Trust me, soon enough you will. AT&T Wins again.
On Sprint, if the iPhone is connected to Wifi internet, then you can browse the web and also speak on the phone at the same time. The other high end phones on Sprint that have 4G can browse on 4G while talking at the same time. The only advantage on ATT is that you can be on strictly 3G and browse and talk at the same time.