I've read this book several years ago. It's about this girl who has a birthmark in the form of a paw print on her side. Then there is this boy who has the same birthmark and later on she discovers she can transform into a puma, I think. More I can't remember.
Could this be "The Gathering" by Kelley Armstrong?
A year later, Maya is getting ready to celebrate her sweet 16. She wants to tattoo her birthmark—a paw-print shape on her hip. She doesn't want it altered in any way, though; she just wants to make it more noticeable. However, she doesn't get it because the tattoo artist's aunt insults her by calling her a witch in Navajo. Later, she invites Rafe, a new bad-boy at school, to her birthday party at Daniel's house. At the party, the teens have a competition on Maya's new rock wall: if she can beat all the guys, they have to add more footholds, but if she loses to even one guy, she has to kiss him.
Rafe tells that skinwalkers are people with special abilities including being able to shift into mountain lions or bears, and the ability to heal and communicate with other animals. He says that Rafe, Maya, and Annie are skinwalkers and that they carry their animal traits with them even in human form. This explains Maya's enhanced hearing and night vision as well as her amazing skill of rehabilitating injured and sick animals. Maya makes the connection between her paw-shaped birthmark and the fact that she is skinwalker. Rafe confirms this idea, saying that paw-print birthmarks indicate skinwalkers and that her and Annie both have them.
The protagonist is a young woman, about 18 or so, who ends up getting chased out of her world, and finds herself in a different one. She does gain the ability to change back and forth into a greater cat - a leopard rather than a puma, but very similar nonetheless. And there is a boy (or, young man) who hunts her because she had scared the nearby village, but reveals that he is also a shape-shifter (of the same kind of cat).
The things that don't fit so well - there is no paw birthmark, but an inherited silver ring with a crouching cat (one she was literally unable to remove due to having worn it so long it would no longer come off) was part of her transformation, possibly marking her out for having the ability to shapeshift and/or part of the reason why she was offered passage to the other world (I think this wasn't explicit, but implied). Also, this is not a novel-length story, but a short story - though it is the title story of a collection, so it might easily be remembered as the name of the book containing it as well.
Valorum's answer may be a better fit, but your question reminded me of this story, so I thought it would be worth offering anyway. I hope you find your story :)