In some states (like Oregon and Virginia) they do, but in a lot of states the regulations for rate structures are flat among all users so when there's a large surge in new demand the utility will build out new capacity and spread the cost of that new capacity to all rate payers with no regard for the fact that the new capacity would not have been needed without the new demand (from data centers). So everyone who was already using the electricity pays the new higher rates along with the new large-load user.
These companies building data centers will often make a lot of PR statements about how they are fine paying the extra cost for extra use while at the same time lobbying behind the scenes to actually avoid that happening and fighting against changes to utility rate structures that would raise their costs. By and large they can't be trusted.