Should I isolate grounds of an isolated DC/DC converter?

If you would connect the two grounds it's not really useful to have an isolated converter in the first place. It would be like two supply voltages on the same circuit, like +5 V for the logic, and +12 V for relays, or something like that. The two power supplies may also only share their grounds, but that way they're not isolated, not even if they would be otherwise floating, like batteries.

Isolation is often for safety reasons, or to avoid ground loops, like Tony says. One reason for using an isolated converter may be to have a floating output so that you can reference it any way you want. If it's a 5 V/ 5V converter for instance, then connecting Vout to the input's ground will give you a -5 V at the output's ground.

So if you had a good reason to have isolation, don't connect the grounds.

The answer depends on whether the output DC power will be used isolated. For example, will it be running a sensor riding on the power line voltage? Does it need to be isolated to avoid ground loops with external equipment? Does it need to be isolated to allow for fully insulated operation with minimal leakage to earth ground, as is often the case for medical devices connected to patients?

If you just want a different DC voltage in the same ground-referenced circuit, then a isolated supply is overkill. In that case you tie the two grounds together to make a non-isolated supply.

THere are as many reasons for isolation and common ground in DC-DC. The issues are tradeoffs but tend to cost a bit more with dual isolated windings vs a single wound magnetic part. Hence, the reasons need to justify the extra cost.

In addition to Olin's reasons to;

a) avoid external ground loops

b) provide additional protection to leakage in medical approved patient apparatus safety

c) Providing common solution for a V+ or a V- out.
- Often referred to as inverted output but with the option to connect ground to + or - output

The down side is floating secondaries are more likely to have high CM noise without a low impedance ground, they are more likely to cause EMC egress to high gain microphone circuits with poor CMMR because the isolation leakage gets works as the harmonic content increases with coupling capacitance interwinding creating some issues with telephone inputs and other analog peripherals in some cases. THis is design specific and not all across the board of DC-DC convertors.