Yes and no, it's hard to say.

Individual things will always be less efficient than mass produced things.

Food

Factory produced items.

Electric vs Gasoline cars.

Local vs central models.

But also, people might use it less frivolously if they are footing the bill directly. Plus, you will still need a data center somewhere in the chain to train the models, you cannot train models on local hardware. Not with any level of meaningful usefulness (AI hate it's never useful blah blah aside).

The main advantages of local AI are two things. First Privacy, which is huge. No stealing your prompts of tracking what you ask about for ads or potential abuse by fascist assholes etc. Second, it's not edited or censored (with the right model). You can ask it whatever you want from Tianamen Square to making bombs, to naked pictures of your favorite celebrity etc. If this is good or not is debatable, but it's one of the two main reasons local is better.

It's probably the case that a data center is better at running an AI than a home PC, they're crazily optimised. I also doubt it would be more cost effective, unless you were using the hardware all day.

Kinda like folding at home.

It depends. A dedicated AI-optimised GPU / NPU would be more efficient than a normal GPU. But it would probably need water cooling, and there would a cost for transmitting signals from your computer to the datacentre and back. If your local grid and the datacentre's grid use different proportions of renewable electricity, that is another factor that will need to be considered.