Instead of dealing with an electrician, might I suggest an alarm technician? You're talking about low voltage wiring and this is what ATs specialize in. Electricians would just as soon strap your network wires to a 240vac cable.
As to cost: a top tech., working "on the side" would probably charge around $80/ outlet incl. wire, outlets, and labor. If your basement is unfinished, it might be less.
The only problem you'll have is finding a tech. that has the time, or inclination, to do it. The networks I have wired are always service hassels. It goes like this: There's a problem on the network, the network gurus (wireless types), always blame it on the wires, wire tech. shows up, spends an hour or two (enough to trash his day), finds nothing wrong with the wires. Meanwhile the gurus are having a "eurecka!" moment. Say nothing. Tech. leaves. System works. Had to be a problem with the wires. Yeah, right. That's why we no longer wire networks (except to "future proof" a new pre wired house), unless we sell and service all the equipment.
As to self installation, attic to basement runs, psiu hit on one good solution with the return air shaft. Use plenium cable if you want to do it "right". Another way is to run beside the vent shaft for the house drains (never liked this one so much). If you have stacked closets run it inside the closets. Or my favorite for a 2 story house, get out the 20' drill bit out and poke it down thru stacked interior walls. There are a lot of little tricks used, just takes a lot of "stick time" to learn to apply them.