Do they mean something like "please go! You must leave!" or could it be "We assure you that he left"?
Noting that @tchrist assures us there can be no ambiguity in AmE, I feel I should post the contrary position. Which I can't substantiate as representing BrE in general, but it's certainly how I see it.
There's no doubt the untensed "infinitive/subjunctive" They insisted [that] he leave only ever means that what they demanded (of him, or whoever controlled his actions) was that he should leave.
But to me at least,...
...is perfectly acceptable. But if it weren't for this question reminding me of a possible (perverse) interpretation, it would never occur to me to think it might mean she insistently claimed later that this is what happened.
In short, I don't think it's particularly uncommon (or generally perceived to be a significant error) to use X insisted that Y Z'ed to mean they were adamant that Y should do Z (was required to, at the time). The alternative interpretation - insistence that Y did in fact do Z - may be equally credible in some contexts, so I would simply say the construction itself is inherently ambiguous.
In American English, there is no ambiguity. There is no issue with it being idiomatic; it is. Your sentence is in the indicative for us:
They insisted that he left.
That version always means that they are assuring you that he has indeed departed. It is a done deal.
Had we meant the other thing, we would have said this, using an untensed verb in the subordinate clause to indicate subjunctive effect:
They insisted that he leave.
Which means that they had demanded his departure. Whether he actually left or not is unstated.