Because you said your country of choice is Canada and everyone else has talked only about the US or Europe, here's a Canada-specific answer – which is, like for many things, kind of "in between" US and Europe. (For context, I'm a professor in computer science working on machine learning, so statistics-adjacent, at a top Canadian research university; I did my education in the US and a postdoc in Europe.)
Canadian master's programs in applied stats / data science come in two flavours, which are very different from one another.
One flavour is a "professional master's." This is more common for things called, say, "master of data science"; these programs are course- and/or project-based, and aimed at getting you a job as a data scientist. They're more often 1 year, sometimes 2. These programs charge substantial tuition (my university's program is C$35,000 for Candian citizens + permanent residents, C$55,000 for others, with some amount of aid/loans available but it's going to be Expensive). They are very industry-focused and do not really focus on preparing you for research, but do prepare you for data scientist-type jobs. (Whether these programs are useful to you might also depend on the particular program, what you did in your undergrad, and what you've been doing in your SWE job since; there's a fair amount of variety in how much they teach you CS things versus specifically data things.) Most US master's programs fall more into this category, though there's a bit of a range.
I am not very familiar with the process for getting into professional master's programs, but you certainly do not need to contact professors for it; you need to convince whatever admissions committee that you're motivated, qualified, and likely to succeed. Talking about your self-study in an application essay, if applicable, may be a very good thing, but you also may have some trouble getting in if there are many prerequisites you don't have official "proof of." It might help to take a course for credit somewhere as a non-degree student; this will be much easier if you've already self-taught the subject! But this is going to vary a lot depending on which particular master's program you're looking at.
The other flavour of master's is a "research master's." Here you take courses, receive a stipend (probably based at least partially on TAing, and the amount varies dramatically across Canadian universities), and as a large component of the program do novel research with a faculty supervisor. These are typically two years, although at least here people often take three. This degree roughly corresponds to the first few years of a US PhD; if things go well on both ends, it's common for people to continue on with a PhD with the same supervisor, but it's also common for people to switch to do PhDs with someone else / at a different university, or to go get a job instead. (There is typically much more research involved than a European master's; although not a strict requirement, the general expectation in my department is that master's students complete at least one project corresponding to a publishable paper in a top venue before they graduate. This is somewhat different in statistics, where publication is slower, but not super dramatically.)
For people applying to research master's in computer science departments (and I think stats is not too different), it is absolutely the expectation that you will be able to talk about research when applying. It's not that you need to have smart things to say about every professor you're interested in's research or that you're expected to know everything already (certainly not), but the general expectation at least for the students I admit is that you should have both a strong coursework background in related areas, and ideally have done a little bit of research of your own already to have a sense of what it's like and that you like doing it. Not having done any research before is not necessarily disqualifying, but you need to stand out in some way from the many applicants who have done that. Different people put differing amounts of weight on this, though, and many don't care at all, just look for people who look smart + motivated. So, if you don't have an immediate path towards doing research (and most people won't)...look smart + motivated.
(Having a specific problem you want to work on is a mixed bag; it can be a good sign that you know what you're talking about, but also the potential advisor might be worried that that's the only thing you're interested in and maybe that doesn't align with what they care about.)
For getting into these programs, pre-existing contact with professors is complicated. If I know who you are and have a positive impression of you, there's a much stronger chance that I'll pay attention to your application. But if you send me a cold email that says "I found your work on [clearly copy-pasted paper title from my website] so inspiring!," I'm not going to reply, if I even open it – there's a flood of those emails at admission time, which are generally clearly extremely low-effort. So...one strategy is to put effort into contacting a few professors, make it immediately obvious from your email that you've made that effort, and even then accept that many of those emails will not be read. Given that that's a lot of effort for low probability of payoff, you can also just not email anyone, and just make your application the strongest you can make it.
In terms of prerequisites: I'm not exactly your target audience, but honestly I would be very leery to admit someone who hadn't taken and done reasonably well in a formal linear algebra course. If it's just that you're refreshing some tools you haven't used in a while, great! But if you've never taken linear algebra / multivariate calculus / probability / ..., getting into a data-oriented Canadian research master's may be difficult, and so might succeeding in that program if you do get in. If you're just missing a bit, you can probably catch up, but if you're missing a lot, it'll make everything much harder.
So, learning these topics is a great strategy, but "I self-studied X topic" on your CV is hard for me to really trust. If something else is drawing me to your application, I could give you a "math test" and have you convince me that you do know these topics – but I'm not going to put that effort in unless there's something else convincing me that doing this is worth my time and effort. So, again, getting course credit for any major missing topics – maybe after you've self-studied – would help with this path too.
