All depends on the teacher. I've gotten putting lessons from some reputable teachers and realized that they know a lot about the golf swing, but stay away from them when it comes to putting. OTOH, I've met some really brilliant minds when it comes to putting.
If you've been on this site for awhile you know that Brian doesn't believe in there being 'one way' to hit a golf ball. I believe the same applies for putting. Nicklaus had a different putting stroke from Tiger who had a different putting stroke from Crenshaw who had a different putting stroke from Loren Roberts who had a different putting stroke than Bobby Locke. All were great putters.
The problem is that most putting instructors or people who give putting lessons teach one particular stroke method and feel things like 'eyes over the ball' are an imperative to putting (which that is definitely not).
I would probably suggest finding an instructor who has the SAM Puttlab. This way it gives you objective information about your putting stroke. The instructor may try to get you to utilize one type of putting method, but at least with the SAM Puttlab data you know what your stroke is doing.
I'd also go and get the putter fit by a clubfitter that has the Mitchell Putter Fitting Studio. The main goal of those things appear to be to reduce the amount of skid in your putts and to get you the proper lie angle. I had a Mizuno Putter that was supposed to be at 71* lie angle and 3* loft. When they measured it, it was actually at 70.5* lie angle and 4.5* loft. The difference in lie angle didn't bother me much, but the loft difference really bothered me. Either way, they wound up fitting my lie angle and found that my lie angle should be at 74*. So getting a putter fitting if anything will at least check to see if your putter is at the lie angle and loft it is supposed to be. And the cool thing is the clubfitter should be able to adjust your own putter for you. I actually wound up getting a new putter since we felt that one with a heavier putterhead w/grooves on the face would help me even more. But if you're strapped for cash and don't want a new putter, they can just adjust your own putter for you.
Getting the putter fitted for me has helped my putting instantly. My short putts are almost automatic now and if I could get a little bit of luck, I would be making a ton of putts outside of 15 feet, but I keep getting some crazy lip-outs for now.
I'll PM you some other thoughts on the subject.
3JACK