Everyone has to make their own decision based on their knowledge and skill level, and the time and money involved.
IMO 100% of riders should be able to change a tire, adjust seat or handlebar, replace brake shoes, and do basic adjustments on their brakes and derailleurs.
The next level includes replacing cables, more subtle derailleur adjustments, servicing bearings, replacing cassettes and other basic assembly and repair work not requiring very expensive specialized tools. If you're a serious rider putting in lots of the miles this is stuff you might want to do, or you might be very happy to let a pro handle it for you since it isn't every day kind of stuff.
The last level, involves expensive specialized tools and skill and knowledge for things like facing the bottom bracket or headset on a frame, frame straightening, building wheels, or other more complex work. The economics rarely favor doing this yourself, but many do either because there's no reliable mechanic locally, or because they enjoy the work and trust themselves more than strangers.
There's no hard and fast rule as to where the line is, but before investing too much into specialized tools, consider that bikes change and you may only have a few years to recoup the outlay vs. paying as you go for service.
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