IME, not at all during the hiring process. And only rarely later.
Your comment below gives one side, where companies want a specific thing and don't want to think beyond that. There is another side to that, where the company can't afford a dedicated engineer for X, and so being able to do some of that work is very valuable. I think this is probably more common in hardware startups more than anywhere else.
I've worked at companies (< 10 people) where we absolutely wanted different (relevant) skills, because we needed people to be reasonably autonomous in carrying out tasks. For example, we wanted people who had optical lab experience, could program and if you had mechanical engineering that was a big bonus.
My suggestion was that the odds of needing multiple skills is larger at a 5 person early stage startup than an established 200 person company. The larger you go, the more you're at the whim of HR screening. There may well have a long list of requirements, but on the job you're probably going to be more specialised.
IMHO, this is mostly due to the nature of the industry I am in (software). Hiring engineers is difficult, and only done when there are specific needs to fill - particularly at small companies, that can't afford to think too far ahead, or afield.
Agreed, although I can also detect a lot of negativity from an older generation of academics in fields that classically border machine learning very closely. One remark I found funny (paraphrasing) is: "Total learning is a constant, and it consists of the sum of the machine's and the person's learning.. the deeper the learning of the machine.. the shallower the learning of the person". But I don't really have a stake in this and am not in much of a position to judge.
This really depends on your employer and your definition of specialist. There are jobs where following a very straight career path would be a disadvantage: you might be perceived as a boring candidate who would potentially struggle if given work outside their domain. I think it's fair to say that, generally, breadth is valued more at smaller companies where people have to multi-task.
In academia it's much easier to do well as an inter-disciplinary researcher than it is to try and be the best in a single field. The obvious killer combo right now is machine learning for X, where X is a field which does not have a strong grounding in computer science. Hacker news gives a very skewed view on the world, as if everyone is up to date with the latest developments in deep learning. In reality, most people in academia are pretty unaware of what's possible, or don't have the technical skills (or grad students with them) to make use of it. This is currently true even for maths-heavy fields like physics, which you may find surprising - I certainly do. Now is a really good time to get on the ladder.
You also need to consider job satisfaction. People who are pretty good at several things may be bored with a single track career, and no amount of money will make up for that.