I'm kind of confused. I thought people who got tattoos wanted people to notice them.
Speaking for myself, my Beloved, my Beloved's sister, my tattoo artist, and a number (though not, by any means, all) of other tattooed people that I know: no
Like PPs have said, a lot of people get tattoos for personal reasons. I currently have a full back tattoo, a cuff tattoo around one ankle (that goes down across my foot also), and a small tattoo on my inner wrist. They're all very personal, I love them all very much, I adore looking at them/knowing that they're there (and I actually shave my legs more because I love the cuff so much, but if I shave one leg it looks weird, soooo...), and if I could make them invisible to the rest of the world (or just to anybody I didn't know well), I would in a heartbeat.
When people say to my tattoo artist "You just have that art so people look at you. Why won't you stop and talk to me about it? What's your problem?" (people often become hostile because she doesn't want to stop going out her business to answer questions about her tattoos for the next half-hour) will say "Sorry, my invisible cloak is at the cleaner's today. Wish I had it, so you couldn't see me".
It can be exhausting to have tattoos exposed and be stopped every ten minutes by someone who wants to ask about them. They also seem to significantly up the "dudes think it's OK to cat-call and say obnoxious, sexist crap" factor, which I hadn't realized would be a thing that happens.
Because I like to wear minimum clothing in the summer (hot!), and would also like a minimum of tattoo comments, I made some HUGE patterned silk gauze scarves this summer. They're light enough (and pack down small enough) that having one draped over me isn't too bad, if I reach the point of "If one more person tries to strike up a lengthy conversation I'm going to cry". The pattern makes it hard to realize that there's art on my skin.
It's not that people nicely complimenting them is bad. It's just...exhausting, honestly. If I see someone with body art that I'd like to compliment, I do the same thing I do for any other compliment, which is first take a good look at them and try to deduce if they're receptive to talk to strangers. If they're reading a book, playing a game, listening to music, or generally look like they're in their own world, I leave them alone. If they look alert & friendly (not tired, grumpy, sick, or unhappy) then I'll say something. If I can't tell how they're doing, I leave them alone.