Just to pick up on this specifically, because its something many people don't realise and it comes up a lot...
No, not necessarily. And - and this is the bigger issue - even if they do, it doesn't follow that it confers a normal (or even any) level of protection.
Someone with a compromised immune system can have far more going on than just an inability to defend against infection. They can also lack the ability to 'learn' new viruses - which means immunisations don't work well, or at all. In fact, if your immune system can't identify the threat and react to it, it isn't going to learn it.
Too, there are illnesses and treatments which wipe ALL acquired immunity, so that it has to build back up from scratch - bone marrow transplant, for example. They can often be rejabbed, but it takes time for the system to come back online, as it were, and they can't all be done at once. I've seen revax schedules for people lasting over 2 years, with real restrictions on their lives while it happens. No hotels, no crowds, no public transport - an outdoor coffee cart in the middle of a walk is exactly the sort of place they would go.
It isn't expected that the general public understand the ins and outs of all that, or even really think about it. Instead, we set reasonable general guidelines to help to act as a compromise. For chicken pox, that's stay away from other people while spotty.