Xbox Live Arcade (XBLA) is really just a promotional name for download-only games that Microsoft releases regularly (generally every week on Wednesday, but sometimes on Friday as well).
I assume the game you are referring to is Tony Hawk's Pro Skater HD. While this game was released as an XBLA game, at the end of the day it's really just an Xbox 360 game that is only available for download from the Xbox Live Marketplace.
All XBLA games come with a free trial. What is available in the trial varies from game to game, but generally you can play the first few levels to get a feel for the game to help you decide if you'd like to purchase the full game or not. Typically the trial version doesn't let you play online, but there are some exceptions to this (such as if the game is an online shooter).
The only other Tony Hawk game found when searching Xbox.com is Tony Hawk's Project 8, which is a traditional disc based game released in 2006. While some disc based games are available for download from the Xbox Live Marketplace, this isn't one of them.
To answer your question, there is only one Tony Hawk game that you can currently download from the Xbox Live Marketplace, and as long as you purchase the full version you should be able to play it online (assuming you have an Xbox Live Gold subscription, of course) with anyone else who has purchased the full version as well.