Monaco has very strict rules about residency. I think you have to live there for 10 years and be asked personally by the royal family to become a perm resident. Maybe they didnt realise.
I don't know the whole rule book, but you become a resident by having X amount in the bank at all times (liquid, ie cash, I think 1mil euros) and (obvs) finding a place to rent (or buy).
Citizenship is another beast, I think for that you need 10 years residency and Al decides personally who's granted citizenship (it's not entirely clear what he bases his decision on aside from the law, as he's refused a number of people who easily met the criteria for citizenship). It will become even more difficult to acquire Monegasque citizenship in the future as citizens have additional privileges and Monaco is pretty full (they are at max capacity for citizens and privileges granted to said citizens aren't going to be sustainable with too many more citizenships). Plus they require you to relinquish any other citizenship you have.
Petra and Sam could easily reside there, I assume Petra has the necessary cash minimum requirement and renting (or even buying) shouldn't be too much of a problem either.
Aside from that I am actually surprised that she bases her family in the US anyway, since the whole family is in Europe and Europe has more to offer in terms of quality of life, geographically being closer to family and so much else. Tbh I am shocked these days that anyone would choose the US over Europe...