Ethernet cable comes in many varieties. For now, let's assume you are talking about Cat5e 4-pair cable. Anyone can run it, but electricians are experts at snaking wires through walls and may be the best (even if not the cheapest) choice. But any contractor or handyman who has some experience doing it can do it safely, as long as they keep away from the regular electric wiring.
Most (I can't say necessarily "all") of the issues with electrical wiring do not apply because the voltage & current are much less dangerous than with household AC wiring. But the tricks & techniques are the same. I gave my electrician a short course in Ethernet cabling years ago because he knew how to run the cable but didn't know anything about the ends.
I highly recommend that the cables be terminated with jacks. I have seen many installations, including by "professional" installers, that terminated with a cable coming out of a whole in the wall and plugging directly into a router or other device. This is in my opinion, unprofessional (who wants a wire coming out of a hole in the wall?) but more importantly if the end of the cable ever goes bad (and it will, the day of a big deadline) you have to crimp on a new connector. Properly installed jacks rarely go bad and if the cable from a jack to a device goes bad you toss it and plug in another one.
Unlike regular electrical wiring, Ethernet cabling can use box eliminators instead of full electrical boxes, though if you have boxes available (e.g., from previous projects but without any live electrical wires inside them), you can use those too. If you have existing old phone wiring that you no longer need, then you MIGHT be able to use it to pull new Ethernet cables through the walls, but keep in mind that telephone wiring (especially residential) is often wired in a daisy-chain fashion while each Ethernet connection will normally need its own home run.