You are correct in your research so far. A rule of thumb is that a shield is always grounded at the signal input ground (The ground of an IC op-amp), not power ground, even though it maybe a ground 'trace'.
A more extreme example would be ultra-sensitive brain-wave sensors. Yes, a common signal ground is attached to both ear lobes, but in addition each input wire is shielded, with the shield connected to an op-amp output equal to the inverse of the input signal. This is an 'active' shield that neutralises all but the most extreme noise.
Connecting a shield wire at both ends allows DC offsets to occur, but can be cancelled out if both drive and receiver are differential. Sometimes this is done with a 100K resistor at the signal source ground and the cable shield connects to the resistor. The idea is to allow a high impedance reference to keep a floating source signal in the common-mode range of the inputs, but not allow offset currents over a few microamps to develop.
You do NOT want to have an ungrounded shield. It would act like an antenna and pull in all kinds of noise. If the wire or twisted pair is shielded, it needs to be grounded as mentioned above.