Obviously, some dedicated powers have use limits—the greater shout 3/day and true resurrection 1/month, for example—but most don't. That means, when an intelligent item has a dedicated power, if the intelligent item believes using the power will further its special purpose, the intelligent item can take a standard action to use the dedicated power. Really, as often as it wants to. The Dungeon Master's Guide on Dedicated Powers says
A dedicated power operates only when an intelligent item is in pursuit of its special purpose. This determination is always made by the item. It should always be easy and straightforward to see how the ends justify the means. That is to say, if the player’s reasoning for how a particular action serves the item’s purpose is not completely believable, the item won’t allow it. Unlike its other powers, an intelligent item can refuse to use its dedicated power even if the owner is dominant.... (270)
Just so we're clear, though, when an intelligent item is introduced to the campaign, the DM has added an NPC to the party with its own personality and agenda, and it's usually a sharply limited personality and agenda. An item with a special purpose—mandatory if the item has a dedicated power—is even more single-minded than most intelligent items yet, for example, has the Intelligence to put forth a well-reasoned argument that all gnomes should die, the Wisdom to know when it's time to go kill some gnomes, and the Charisma to convince its wielder that the whole gnome-killing thing is awesome—pick two! These are semi-creatures with two mental stats of at least 17 and the third at 10 that will explain these things to the wielder all the time telepathically. So if the wielder wants Gnomoregnomes to shut up and use its awesome dedicated power, the wielder better get with the gnome-killing.
And if the intelligent item doesn't think that using the dedicated power will further the special purpose, the item just won't employ its dedicated power. In most cases, an intelligent item with a special purpose and dedicated power can afford to play the long game.
The intelligent item is functionally immortal. If the wielder dies or loses possession of the item because the wielder wasn't pursuing the item's purpose, maybe the next poor sap who wields it will be more open to, for example, killing all the gnomes, y'know?
The intelligent item is extremely valuable. If the wielder dies or loses possession of the item because he's halfheartedly going along with the item's goals—like, the wielder killing the occasional gnome of opportunity but really offing kobolds by the tribe—, the item is either smart enough or wise enough or both to know that being captured by its hated foes means it'll probably be sold by those foes to a buyer far away from that group of foes but near enough to a different group of identical foes because that'll be where the item's most valuable. (Remember that any intelligent magic item with a special purpose is worth at least 63,300 gp without adding the cost of the actual magic item! That's a lot of cash to pass up just because a sword hates you.) For example, extremely practical surface elves would probably be totally comfortable selling a magic item with the special purpose slay elves to svirfneblin who will then use the weapon against the drow.
"We're surrounded by giants! My sword hates giants! Cool!"
Probably. Remember those mental ability scores? If a creature that's wielding a magic item that has a special purpose to slay giants is beset by the giants in the middle of Giantville, although the item might be happy to slay peon giants for a while—especially if it's been some time since the last giant-slaying—, eventually the magic item will no longer be content slaying just any giant. No, sooner or later, the item'll probably want the wielder to go after the giant tribal leader (or, in the case of storm giants, the head of the family). And if the wielder insists on instead murdering the giants' minions or slaughtering noncombatant giants, the item's likely to lecture the wielder (telepathically) about how it's supposed to be employed then teach the wielder a lesson by not using its dedicated power unless worthier foes are at least promised if not delivered.
For consideration only—not an excuse to be a jerk
It sounds like there might be some concern about the intelligent item's dedicated power being abused. If the PC's making a good faith effort to satiate the intelligent item's bloodlust (or whatever its special purpose), then the item should routinely and reliably make use of its dedicated power on the PC's behalf when the PC's furthering that special purpose. Unless the item was randomly rolled, the DM's introduced to the campaign a powerful magic item with a big neon sign on it that says You should go have adventures doing this and if the PC is doing that, the reward is the intelligent item using its dedicated power for the PC's benefit.
So the above perspective is for when things are getting out of hand. I can't know what out of hand is in other groups, but in mine it would include lying to the intelligent item to get it to use its dedicated power at inappropriate times or threatening the item in an effort to get it to use its dedicated power. Both should have repercussions, just like lying to or threatening any NPC does. Velma doesn't lie to Scooby-Doo about rewarding him with Scooby Snacks, and Fred never says to Scooby-Doo, "Listen, dog, help us catch the amusement park phantom or I'll sock you!" Treat the smart, wise, personable magic item with the same respect.