It's older than you might think.
Versing: The Video Game Connection
Anecdotal evidence seems to point to "versing" making its way to the mainstream through video games that came out in the '90s.
Ben Zimmer noted in a post to an American Dialect Society discussion group that you could find comments that used "versing" to mean "playing" on gaming newsgroups back in the mid-1990s. For example, one gamer offered this advice on a Sony video game list in 1995: "When versing the black car, remember that the first is a warmup lap..." A couple of my Facebook commenters noted that their kids started saying "versing" when they started playing Pokemon, and Pokemon was created in the '90s.
The first entry for "versing" showed up on the Urban Dictionary site in 2004 and reads "Derived from the common term 'vs.' in video games where choices are either 1 Player or 2 Player (commonly listed as 'vs.')." Martie on my Facebook page said her boys use it and "I tried to correct them, but learned it's the gaming lingo."
My theory is that either kids were exposed to phrases like "A versus B" much more often because of video games, and they started making the verb mistake themselves more often and in groups where it got reinforced, or that some of the games that came out in the ‘90s actually used "versing" as a verb. Either way, it spread, became more accepted, and eventually also made its way into sports.
At this point, "versing" falls into the category of slang. It seems similar to "woot," which also has roots in gaming. Older people might not have heard it before, but most kids know what it means. It has a place in niches like gaming and maybe even sports in some regions or countries, but it would seem wrong to most adult Americans if they saw it in USA Today or on CNN.
Versing: The Old Meaning
"Versing" actually was a verb about poetry before kids started using it the way they do today. The Oxford English Dictionary has it being used as far back as the year 10001 to mean "to compose or make verses." My favorite example is a sentence from 1909: "He began to verse extemporaneously in her ear."
Other Video Game Terms: Pause
"Versing" isn't the only gaming term that has made it into real life. Two people reported that their kids say "pause" or "pause game" when they need a break when they're playing outside.
Versing: Are We Reversing?
Finally, I'll leave you with this joke from Janice C. She asked, "[If kids say 'versing' to mean playing another team, If they play them again, are they reversing?"
If you or your kids play a game that actually uses "versing" as a verb, if the game says something like "You are now versing player 2," please leave a comment and let me know.
Ben Zimmer e-mailed after this article was published to point me to a later post to the American Dialect Society group that I had missed: In 1984, the New York Times noted the word "versing" and called it New York "high school slang meaning to compete against another school's team."
1984 is much earlier than the other examples, and one would think "versing" must have been quite widespread to merit a mention in the Times (although it could also just be that the writer's kids and friends were using it a a lot).
Although two anecdotes aren't enough to let us draw meaningful conclusions, the Times mention combined with Bob P.'s story about his sister using "versing" around the same time in New York makes me wonder if "versing" got a regional start in New York. It could have spread from there or arisen elsewhere independently for the reasons I cited in the original article above.
Another Update (2019): Ben has pointed me to an even earlier example from 1981.
1. Note: I said "the 1600s" in the audio podcast, but when I double-checked later I found much older examples.