Energy efficient, yes, but when you want to keep a query/data private it's maybe worth the extra $ KW. Chicken pie recipes and google AI-search can still go though the datacentres.
As an aside: The computation might also not be the same e.g. ever-changing hidden pre-prompts, security/safety checks blocking or degrading responses, unavoidable verbosity to simple questions, watermarking, collection of prompt data to build user profiles for the purpose of advertising - and we haven't even seen in-response adverts, or sponsor-biased responses yet, but no doubt it's coming.
I want that too, but you gotta ask yourself the question how efficient that is compared to running it in a datacenter shared with everybody else.