Are you referring to shielded Network or ADSL cable ?
Basically replacing a flat profile (non-twisted pair) telephone cable with one that has twisted pairs offers a significant improvement to interference rejection, shielding however will have minimal impact on the connection unless the shield is grounded at one end.
Network cables similarly will need to be grounded and your network card and router are unlikely to have connectors that will have a ground connection so there would be little to be gained in replacing a standard CAT5e cable with a shielded version.
I believe shielded cable does work well, even if it's not grounded. The reason to ground the shield is to prevent the build-up of a dangerous charge if the cable is carrying high power. But like you say, it's important to only ground one end if it is grounded, to prevent a hum-loop.