I'm trying to establish a data connection with my GSM socket modem. The remote modem answers but the laptop still makes a beeping sound of waiting for connection and if I type a key on the laptop it will log out with the message 'NO CARRIER'.

Make sure you have data service. Check with your carrier and ask for circuit switched data (CSD). Some carriers do not offer this service. With CSD you should be able to originate a data call from the mobile modem to a land line modem. Your maximum speed will be 9600 or 14400 (V.32 or V.32bis).

If you also want to be able to answer a data call with your mobile modem, then you need to get a second phone number from your carrier for incoming data calls.