Are they actually lightsabers? Did she get them (or the whole weapon) from scavenging?
The Visual Dictionary says "no", or rather doesn't say "yes".
She did scavenge the quarterstaff but it seems to be a regular normal one, NOT lightsabers - at least I'm pretty sure that if it was, it would have been stated in the Dictionary or the film or the novelization.
Also, to address the fact that it looks too elaborate to be a simple staff - it wasn't "simple" stick:
A multifunction staff strapped to her back made precision work in such tight and dangerous quarters difficult. (novelization)
Another point indicating it wasn't anything outlandish: staffs apparently weren't uncommon on Jakku, Plutt's flunkies had them too:
Coming up beside her, one of Unkar Plutt’s assistants barked at her and gestured in her direction with his staff, implying it would be in her best interests to focus on her work and not allow herself to be distracted.
Rey is very mechanically talented (source: Visual Dictionary). If her quarterstaff was made of lightsabers, I suspect she'd have figured it out by hacking around with it.
DISCLAIMER: there's of course an outside chance that it has hidden lightsabers that nobody including Rey knows about and is being held as a surprise for Episode 8. But if so, of course we would have absolutely zero canon information proving or disproving it till then
I could argue that nobody would just leave around a Lightsaber Quarterstaff on a planet that Rey just happened to end up on... but then they left the Map, then BB-8, then Force-using Finn, on that planet too, so I'm not surprised if we find out someone also left a cloning facility, a frozen Dark Jedi, Vader's secret notebook and an abandoned Journal of the Whills there while at it.
Some evidence FOR it:
According to this forum:
Apparently on the Google Star Wars page, the image file of Rey's staff is labeled as "lightsaber.png" in the filename. Considering this was probably put together by official marketing people from Disney (or at least the art assets would have come from them) you have to wonder.... Or it could be someone using a popular search term in the filename to help with SEO.