As others have commented, "it depends".
How a shield functions is a really "exciting" topic, with much that is counter-intuitive. The behaviour varies hugely according to the frequencies you are considering, partly as a result of the frequency impact on impedance, and partly because of the skin effect.
I have learnt this through a mix of physics, and painful lessons from equipment with coax connectors that wouldn't reliably pass EMC certification...
To illustrate, consider:
Two battery powered circuits, in metal boxes, connected with a single piece of coax, metal BNC connectors at both end, with the barrels directly connected to the metal cases via the mounting nuts. There are no other connections from the boxes, and no "ground reference".
The question is:
If one box is driving a 1MHz signal along the centre conductor, where do the return currents flow, and does the "shield" of the coax, actually provide shielding?
I've seen people argue that these is no shielding because of the lack of a ground connection. I've also seen people argue that the "shield" isn't a shield, because it also provides the return path for the signal on the centre wire.
In reality, if you take an EMC tester to this setup, you will discover the shield is working very well, and despite the coax shield carrying the return current, it is virtually impossible to detect any 1MHz bleed. Similarly, the circuits in the box will see virtually no coupling from even strong external RFI sources.
Well at 1MHz the "skin depth" (how far an electrical signal penetrates into a conductor) for copper is ~65um. The result is that all the return current for the 1MHz signal on the center conductor is flowing in a thin layer on the inside of the shield, and all the RFI is flowing in a thin layer on the outside. The story at the BNC connectors is similar, with the RFI being diverted to the outside of the metal boxes, and the 1MHz return currents flowing on the inside to the ground pins on the PCB.
Effectively your circuitry is inside a Faraday cage.
What about if you ground the boxes at both ends?
Will some of the 1MHz return current now go via the large external loop, down one ground wire, through the external ground, and back up the other ground wire?
In practice the 1MHz return current will still flow on the inside of the coax, because the inductance of the loop makes the impedance hugely higher, and because the skin effect and metal connectors means the signal simply can't pass to the outside of the box.
What about the "ground loop"?
At lower frequencies (say mains at 50/60Hz) you have indeed now created a potentially very effective magnetic coupling loop, and if there are large varying magnetic fields near by, these can cause significant currents to flow in both the shield and centre wire of the coax. Depending on the precise transmitter and receiver topology, these currents could cause problems, especially if you replaced the single coax between the boxes with two or more.
This is the classic "hum on HiFi separates" problem.
What's the solution?
One solution is to avoid multiple grounds.
Another is to design the transmitters to be resistant to common-mode currents.
Putting aside the use of optical links, the ultimate tolerance of common-mode currents is probably to use shielded twisted pair, with isolated differential transmitters and receivers. This avoids return currents in the shield, isolates the signal wires from the external ground, and the twisting reduces the loop area of the signal path to almost nothing.
What about if you want to keep using coax, and "single-ended" signals?
The standard solution here is to use insulated BNC sockets, with a decoupling capacitor as close as possible between the barrel and the front panel. In fact you can buy "decoupled BNC" connectors where the capacitor is built directly into the connector.
In this case, the circuitry inside the boxes (and the shield) are floating (i.e. not connected to ground) so no common mode current flows due to ground loops. However the capacitors mean the RFI flowing in the external skin of the coax is still diverted to the metal cases at each end. This works both with and without ground connections to the boxes at each end.
You could choose to use conductive BNCs at one end, and insulated at the other (thereby grounding the shield), but once you have a more than two boxes, or if you need to mix-and-match kits, the potential for loops is too high, so the norm is to just use insulated connectors by default if ground loops might be an issue.
Shielding against LF vs HF interference
LF interference is almost all H-field based, as the antenna lengths required to emit significant EM radiation is huge, and the voltages mean the E-field is limited. HF interference is almost all radiation based, as the wavelengths are similar to the wire lengths.
Metal boxes protect against HF RFI due to the skin effect, but if you have significant LF magnetic fields, you may need to worry about loop area even inside a metal case, as LF H-field can still penetrate...
In the case of the specific question?
If you consider this setup as an adaptation of the metal box/coax case, you can see that the skin effect will still mean that external RFI won't penetrate the shield.
However, the lack of metal boxes at either end, mean that the PCBs and control wires are not in a Faraday cage, and the RFI can leak round the ends of the shield, flow back on the inside, and capacitively couple to the circuitry and wires. Depending on which signal wires carry the bulk of the RFI current, this may cause issues. The best approach therefore is to provide a direct, low impedance path for the RFI in the shield to the system's common DC negative, by connecting the shield to the DC negative at the head end. This will minimize capacitive pickup on the long signal wire, and should reduce the risk of problems.
Similarly, internally generated noise on the signal wires will tend to flow back on the inside of the shield to the negative supply, limiting emitted RFI.
As with the metal boxes, this protection is pretty much independent of whether the DC supply is grounded.
However, this provides much less EMC shielding than if both ends were in metal boxes, and if you are producing a commercial system, you will need to be very careful of internal return paths to avoid hitting EMC issues.
Why have I spent so much time learning about shielding?
On a periodic retest of some very expensive telephony kit I had designed, we discovered the system was failing EMC testing. After much investigation, it turned out that our manufacturing group had swapped the supplier of decoupled BNC connectors, when the first manufacture stopped producing them. The parts I'd originally spec'ed has nice spring connections to the SMT caps which were providing the decoupling. The alternative, lacked the springs and simply replied on assembly pressure. The result was that the weight of the cable connected to the BNC, could cause the capacitors to disconnect and break our RFI shielding.
As removing the BNCs from the already produced kit was too disruptive, we produced a workaround which involved soldering axial caps from the insulated ground pin to the spring washer on the mounting ring. This reliably restored the RFI bypass path, and 100% solved the issue.