For the duration of the fight, creatures in the immediate area will not be visible to you (but they may still attack you if they wander too close). Keep in mind that other players will still be able to see you and attack you if you are eligible for player vs. player (PVP). Other players will also be able to view your pet battle in real time, including the pets involved as well as their health as the match progresses.
Each battle that you win will yield experience for each pet that participated. However, deceased companions will not receive any experience (so make an effort to keep them alive by swaping in a different pet). Experience gains are based on your pet's level compared to the wild pet. Defeating a higher level opponent will grant more experience than a much lower level one, but be ready for a tough fight!