As a first thought, and as something that is often advocated, grounding a shield at both ends may result in severe ground loop currents which could adversely impact EMI and isolation properties.
As a second thought, with the shield grounded only at the output end, if the R1C1 and the R2C2 time constants differ even slightly from each other, the interground interference signal, Enoise, can induce a differential noise signal between the two outputs E1 and E2 that feed the differential amplifier, A2.
As a third thought, grounding the shield only at the input end averts both the ground loop problem and the time constant mismatch problem. Even if R1C1 and R2C2 are markedly different, all of the parts from A1, R1 and R2, C1 and C2 and the shield braid itself share Enoise as a common mode signal so that no differential voltage is created between E1 and E2. The A2 differential amplifier is thereby protected from Enoise.