Machine pistols have been sold for the purpose of being personal protection weapons for people who would only be lightly trained on the use of a handgun. Spray and pray is all you're going to get out of the user anyway.
Machine pistols require far MORE training to use compared to a standard pistol. They are downright dangerous to use without proper training, both for the user and the people around them.
I don't know where to begin on this other than to say handling a full-auto handgun is far worse for untrained personnel than a semi-auto handgun. It's even a challenge for highly-trained personnel.
Additionally, the very long history of machine pistols would indicate the form-factor is a poor fit for the application of any full-auto fire.
This is the primary reason that personal defense weapons (PDWs) were developed in the first place.
I could see that as being a useful role for a VIP protection team where you might not be able to carry larger guns for whatever reason but still want to designate some team members to suppress a potential attacker
Full auto is just going to run through your ammo at the expense of accuracy, reliability, and maintenance time and costs. Nobody is going to be providing covering fire with a fully-auto machine pistol, the ammo capacity just isn't big enough (and then think about cooling and mechanism reliability when putting more than a dozen through a handgun). These things are for raids and assassinations, where collateral damage isn't a big deal but taking out the target is.
That's nice "just so" theory, but is contradicted by the reality that the US Secret Service has been known to use concealed Uzi's, and presumably similar compact full auto weapons, in bodyguard roles.
someguydave is correct. Compact automatic weapons make sense for highly trained body guards protecting VIPs when discretion is considered important.
Right. My understanding is that military doctrine is generally to immediately attack when encountering an ambush. Presumably, that will throw your attackers off their pre-planned attack and help you regain the initiative.
So you want a big enough defending team such that you can immediately assault the attackers while also retreating with your VIP simultaneously. For the counter-assault team, you want to suppress the ambushers as quickly as possible (get their heads down), thus the automatic weapons.
As int_19h points out, there are special-purpose weapons made for this (see "personal defense weapons") and they are likely what pros carry.
Compact automatic weapons still usually have either a stock (even the smallest Uzis do), or some other way to stabilize the gun while firing - e.g. the sling is used for this purpose with some MP5K variants.
IMO the most compelling machine pistols are those with small light weight folding stocks, not entirely unlike what what the Uzi had. Machine pistols could only be the optimal weapon if anything bigger wasn't an option, but my main point is that automatic weapons are considered relevant to VIP protection by the trained experts, contrary to the musings of internet commenters.