As many of you are aware.... the elusive Republic player can often times be harder to find than a legendary artifact making you even question whether they exist at all. As someone who has leveled to 50 on a standard server, I can't guarantee you will ever find one but I can give you a few words of advice to increase your chances of spotting one.
1) Do not be fooled when you suddenly see a red enemy firing at some random object, this most likely is a computer controlled character as this guide would be completely useless if they were that easy to spot.
2) You will find out as you level that often times Bioware has completely segregated Republic and Sith quests making it so that even if they are there, questing alone will not guarantee a spotting. Your best bet is to start exploring all the parts of the map your quests tell you not to go to.
3) Once you locate a Republic base, continue looking until you find at least 3 more, the elusive Republic player will often times only be near of these and even a main base is no guarantee.
4) After locating 4 Republic bases, park your mount, wait 15 minutes then move onto the next one. While your odds aren't perfect you still have a 25 percent chance, which is about the same as winning a piece of loot in a 4 man group, and significantly greater than your odds earlier which were on par with being struck by lightning.
5) BE WARNED Republic Players are very skiddish and often easily intimidated by Sith players, if you do happen to spot one and attempt to make contact make sure you and or your hunting partners pick an appropriate location where you can easily cut off their escape. ALSO BE WARNED, it is best to limit your group size to 2, any larger and they will often times forego questing on that planet altogether (THINK ILLUM).
Anyways Happy Hunting......