Regardless of what the tabloids would think, just because you qualify to be on sick leave for one job does not make you incapable of doing another.
As well as being one reason why most 'critical illness' insurance policies are crap - to claim, they require you to be incapable of doing any job - it means that someone who is legitimately too unwell to do the work of, say, a police officer may well be physically and mentally able to do something else.
On the other hand, the contract of employment for the police doubtless has Quite A Lot to say about clearing other employment with superiors first, at the very least.