Screen Wire

It depends.
Field instrument wiring typically grounds the shield at the controller end
Profibus DP grounds both ends of the shield.
I'm an instrument guy so I ground the shield at one end only.

1) Field Instrument Grounding

from Control System Power and Grounding Better Practice, by David Brown, David Harrold, and Roger Hope

Pg 4-13
Shield Grounds
Proper field-wiring shield grounds help ensure proper system operation by
reducing electromagnetic interference in signal wiring. A proper shield ground
path is to ground the shield at one end only. Grounding at both ends can cause
ground loops. The preferred end is at the carrier shield bar. If field transmitters or
final control elements are well-grounded four-wire units, shield wires can be
grounded to the units instead of the carrier shield bar.

2) Profibus DP
The quote below is directly from the Profibus DP Installation Guide from Profibus.com. It's an RS-485 network.

PROFIBUS Technical Guideline Installation Guideline for PROFIBUS-DP/FMS
Version 1.0, September 1998

3.1 Laying of copper cable
The following cable laying instructions apply to shielded twisted pair cables. The cable shield is used to improve the immunity to electromagnetic interference. PROFIBUS cables should be shielded with a combination of both conductive braid and conductive foil. In the following, "shield" refers to both types of shields (braid and foil). A foil shield should not be used on its own, because it is very thin and can easily be damaged.

The cable shield must be connected to the functional ground at both ends of the cable by making a large-area connection to a grounded conductive surface. When laying the bus cables, particular care should be taken to ensure that the cable shield (braid shield and foil shield, if available) is connected to the shield grounding clamp over a large area.

http://www.profibus.com/download/installation-guide/