It's important enough, and currently wastes enough of my time (read: SEO crap, Medium and Reddit are top results for... everything) that I'd happily pay for a good search engine!
P2P can be done on mobile but you're fighting against limited resources (storage, bandwidth, CPU). The reason P2P got good on PC is because storage got cheaper, bandwidth was increasing and CPUs kept getting faster.
Right. So texting on P2P is easy and appropriate, but I can’t see how building a search engine on P2P would really work well given those limited resources.