According to the wiki, the parent organization of X-COM was US-based. As such, their initial personnel most likely was simply transferred from the US military.
The Bureau of Strategic Emergency Command, or the Bureau, was created
by President John F. Kennedy during the Cold War. Its original mission
directive was to sustain continuity of operations in the event of a
Soviet invasion and entrenched occupation on American soil. It was
intended to provide effective "command and control" to remnants of the
United States Armed Forces, and if necessary, coordinate them into an
effective insurgency.
Later, we know that at least some of the recruitment base for soldiers probably comes from the main body of armies, not merely special forces.
There was no X-COM base where the game wasn’t played. “Where there is
no Crud,” the saying went, “there is no life.” Some went so far as to
claim that X-COM people had invented it, even though it had actually
been caught, rather like athlete’s foot, from fighter pilots formerly
in the British and Canadian forces.
—X-COM: UFO Defense
While they could be speaking specifically about special forces fighter pilots, it seems more likely that they simply mean skilled fighter pilots.
At least when it comes to scientists, X-COM covertly recruits those who have the appropriate skills or experience:
As they passed through the second, heavier set of containment doors,
the ones that separated the alien containment unit from those labs
where only corpses or tissue were held, Jonelle wondered which of the
two categories Trenchard fit in best. His history was mostly
unremarkable except for his involvement in a terror raid, during which
he came close to being killed. Shortly thereafter, he had been
recruited covertly by X-COM, under cover of a shell organization that
claimed to be doing “nonaggressive” work on the alien genome series.
—X-COM: UFO Defense
"Terror raid" refers to the terrorist attacks perpetrated by the aliens against Earth civilians. So Trenchard was a scientist who happened to have encountered aliens and not died, which apparently made him a good choice for X-COM. It does not seem improbable that they do this with soldiers as well.
They have more criteria than "experience with aliens," though.
Lack of major psychological issues is a requirement:
"Colonel. Let me get very straight about this with you. I don’t care
two farts in a high wind about the details of your personal
psychology. It must be at least tolerable, otherwise they would never
have let you into X-COM."
—X-COM: UFO Defense
