From a practical perspective, it's best to shop around for a decent dojo and settle down for a good long while when you find one. After you have a decent grounding in your art of choice, you can go on to cross-train or maybe even take up a second art altogether. It makes you a more rounded and competent fighter, and it also exposes you to new ideas that will allow you to improve your existing skills.
From an ethical position, you should find out whether or not your dojo approves of cross training. Of course, if your doing as you should and staying around for the long haul, the dojo mentality should be obvious to you by the time cross-training even becomes an issue. I know students who have left Korean dojos and owe their lives to the fact that their instructor was a raging alcoholic and couldn't put the kick where it was supposed to go (not to cast aspersions on said instructors skills -he was amazing- he was just a drunken jackass).
Personally, if it were my decision alone, I could care less where people under me train as long as they do what they're supposed to during class, it makes no difference to me whether they go to another dojo, go the strip club, or pour melted chocolate on their nipples and let their cat lick it off. If however, you attend a dojo that espouses Tonys point of view, then abide by the rules set down. If you want to train somewhere else bad enough, straight up quit and start anew.
Iain Richardson, compulsive post-having cake eater-wanter.
"He shoots first who laughs last."
- Alexsandr Lebed,