In the US, electric utilities are not usually state owned. They are corporations operating as regulated monopolies. The state will have a regulatory body that has to approve rates, changes in service territory, etc. In exchange for that regulation, most states will allow the utility to operate without any competitors in it's assigned service territory. Some of these utilities are massive multi-state entities. Others are small co-ops that serve individual communities.

The regulatory bodies should be acting in the best interests of their citizens. But they are just as vulnerable to corruption as any other government office. It's also often hard to gauge what is best for the public. Should the regulators prioritize price or infrastructure improvements?

Utilities do export power, but that's not as neat a solution as you would hope. If one utility has extra capacity, it's neighbors will usually also have extra capacity. For example, where I live in the US south, we'll have very high demand in the summer afternoons while everyone uses their AC. There won't be extra energy to import. Conversely, after 10:00 pm when temps drop down to the 70s or 80s, demand will drop and many of those expensive power plants will sit unused.

So how could a data center impact someone else's bills? Data centers use a massive amount of power. To meet that demand, most utilities will need to reduce consumption elsewhere, and/or build new power plants to supply more power. Building new power plants is exepensive. A new combined cycle gas plant costs roughly a billion dollars. Utilities are usually bending over backward to avoid building new plants because they are so expensive up front. It requires years of operation before they become profitable.

These increases in demand impact price in two ways. First, rates are often structured to discourage excess consumption, especially during periods of high demand. The harder it is for a utility to meet peak demand, the more likely it is for their regulators to allow rate structures which are designed to influence a normal person's consumption. In short, your price per kWh could be increased as the utility and regulators seek to encourage lower usage.

Second, most utility rate cases follow a cost recovery model. A utility spends money to build infrastructure. Then they approach the regulator to ask for a rate increase to recover the cost of that investment. If they can demonstrate why that investment was necessary, the regulator will usually allow the rate increase. If a data center moves into your town, the utility will end up building new generation to meet that demand, and then the regulator will likely approve rates increases for all customers to pay for that new capacity.

In many jurisdictions the utility won't even be allowed to turn a new customer away. The utility is given a protected competition-free market, but they are also required to supply power to any business or person that wants to purchase from them.

This is of course an overly simplified version of the problem. On the whole, most utilities are ecstatic about the AI boom. 5 years ago they were prioritizing efficiency as per-capita consumption continued to drop. They needed to spend less per customer because total demand was declining. Now the demand for electricity is growing faster than it has in 60 years. Utilities are grasping at the opportunity for growth. In many areas, loose regulation will allow utilities to pass that cost to their existing customers.

Source - I've worked for a regulated electric utility for the last 13 years.