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I've been thinking about doing just that. I've been looking at agglutinating languages like Korean and Hungarian. Problem is, I've never managed to truly learn a language unless I had a use for it. I've only ever learned German well, because I used to listen to NDH metal all the time.

I would really like to get a deeper understanding of how these languages work, but I don't see myself being able to actually learn a language that I have absolutely no use for. I can't even complete a German course on Duolingo because that site's lessons are the only thing I ever use German for anymore! I never speak it or write it, and I rarely ever hear or see it anymore.

Learning languages can increase your linguistic knowledge. German, for instance, helped me to understand how a case system is used. I don't know if I could ever have understood a Latin-like case system if I didn't know German.

Do professional linguists do stuff like this? Note that I don't have a degree, I'm entirely self-taught. I've been studying all I can on linguistics for over a decade now, but I've mostly focused on phonology (I got into linguistics so I could make an auxlang, I've since given up on that goal). Recently I've been trying to focus more on grammar, but like I said, I don't see myself being able to truly understand some things when I can't see examples of how its used.

The answer depends on what you mean by "learning" the language.

It's a common misconception that all linguists are fluent in a great number of languages, in the sense of being able to read or speak with native-like proficiency. While some linguists can speak five languages fluently, many others can't, and some highly-respected linguists speak only a single language with any fluency.

However, linguists do tend to study many other languages that they aren't fluent in: investigating certain aspects in greater detail, such as the phonetics of the nasal vowels, or the morphology of the noun class system, or even certain metaphors in the semantics.

For example, my current work is on the morphology of Lingála. I could tell you all sorts of details about the noun class/gender system or how verbs are "extended" to adjust their meaning. But if you dropped me into the middle of the Congo, I couldn't even find a bathroom on my own.

While learning additional languages can be helpful for a linguist, most linguists don't take all the languages they study to the point of fluency. Instead, they learn about the specific aspects they're interested in working with, and don't pay much attention to the others.

In my experience, it is not common for professional linguists to study languages in the sense of general language acquisition study (as opposed to focused technical research). By "study" I mean "study with some diligence" equivalent to let's say equivalent to 75 classroom hours, as opposed to "show up for a couple of weeks and give up". By "professional linguist", I mean a person who makes (made) his living in linguistics -- not translation or language teaching, but scientific linguistics. For example, a linguistics professor taking Saami language classes so that he can speak Saami (for whatever reason). Reviewing the known behavior of professional colleagues that I know, the dominant form of language-learning type behavior by linguists is "technical field working", where one studies the structure of Bilin by working with a speaker of Bilin. This is sufficiently unlike general language learning that it's outside the scope of what I am making claims about (and I suspect outside the scope of the OPs interest).

The second most common form of general language-learning by linguists that I am aware of is "tool language" acquisition where you learn one language in order to study another, for example learning Portuguese so that you can do field work in Mozambique. Before becoming a professional linguist (before completing the dissertation and getting that job), it's reasonably common for students of linguistics to acquire research-tool languages, and this often defines the course of postgraduate research for that person. Once you're working as a full time linguist, it's hard to get the spare time to take a language class, or pursue some self-teaching program. I know only a few people who did that, and in fact I only studied 4 languages qua general speaking languages in the post-dissertation phase (3 of them as "tool" languages). Even rarer are those who study a language purely as the direct object of linguistic interest, for example learning Arabic because you want to understand the structure of Arabic.

There is a third category of language self-teaching of interest, namely the person who focuses on a single language to the point that they gain general ability to speak the language from elicitation, and can start to elicit in the language. Academic linguists sometimes do this; missionary linguists do this quite frequently.