So what happens? Does the Skill get Increased by +10? Or do you get a free skill or are you just left without anything extra and only one instance of the skill?

As far as my knowledge goes, you are just left with nothing extra if you gain a skill from two different sources. It would not, however, be unreasonable to have them stack up as a House Rule. Means that an Operator from a Forge World is probably going to be better out of the gate than an Operator from a Fortress World at Tech-Use, which seems pretty fluffy to me.

I confess I can't recall the specific rule in OW, but in Dark Heresy etc, gaining multiple instances of the same Skill always resulted in them stacking in the form of another +10. I simply assumed it would be the same for Only War.

Page 41. Only War Core Rulebook. The answer is that yes, they stack.

As far as I can read there is nothing mentioned in the Core book about this, and it is also not adressed in the Errata.

*Edit* And apperently i cant read...