One roofer told me the old shingles have to be removed before putting on new shingles. The other said the new shingles can go right over the old ones. Who's right?

If both are competent roofers - a big if - you may just be facing an honest difference of opinion. The most common practice is to put a second roof right over the first and then tear everything off before the third roof goes on. That means a maximum of two layers. However, a second layer shouldn't go over the first if the first isn't flat. If the first layer has curled and broken shingles, it will need to be removed before another layer goes on. One advantage of removing the old shingles is that it provides an ideal opportunity to inspect the plywood or boards under the roof and replace any pieces that may have deteriorated.