This is Ken Bednasz from Telit. You might want to contact Jasper to understand how your SIM account is setup for SMS. I am not sure if you are sending your SMS to the Jasper short code or if you are trying to SMS to another mobile device. Also for receiving SMS you might need to send the SMS from the Jasper control center. It would be best to contact Jasper support team
You can also email TS-NorthAmerica@telit.com for technical support. It might be good to review the AT commands you are using (ex. AT command log).
SMS internetworking is not enabled by default on the AT&T Control Center DevKits - your target would need to also be a device also using a SIM on an AT&T Control Center account - it's not clear from your post if this is the case.
By the way, yes you can send SMS from the Control Center Web UI to a device using a SIM on a Control Center account. There are several ways to do this; one is via the "SMS Live" tab under "SMS" under Details for a device (click on the device ICCID for details). Another is to go to the Devices list, click on the wrench & hammer icon under Diagnostics column on the row of a particular device, then under "Diagnostic Wizard" choose "Send an SMS to the device."
For your reference, here are some previous posts that address SMS internetworking:
Hope it helps,