There is a lot of incorrect information on this thread.
Bacteria lives inside the live rock. Live rock can be moved from one aquarium to another, and if it is kept moist the bacteria will generally survive and be capable of processing ammonia and nitrite immediately upon being placed in the aquarium.
The concern with live rock is not cycling. It is curing. These are different things. Curing is the process of allowing organisms on the live rock to die. These are life forms which will not live in an aquarium, or have been damaged in shipping. When live rock is received at the distributor or LFS, it is generally scrubbed of anything that is dead or dying. As the die off occurs, the bacteria inside the live rock (which are already present) will process ammonia and nitrite, causing high levels of Nitrate in the curing container. The protein skimmer runs on overload, and needs to be emptied every few hours. This happens behind the sceens of retail, and most hobbyists are buying live rock that is fully cured and ready to be placed into an aquarium.
Anyone who has ever set up an exhibit at their state or county fair understands that live rock is a movable source of bacteria. I have set up dozens of marine aquariums and fully stocked them with fish on the same day. Ammonia and nitrite spikes rarely occur. This is no different than moving a fully cycled sponge filter from one freshwater aquarium to another. The needed bacteria do NOT live in the water. They live on the filter, and in a marine system the bacteria is initially in the live rock, and later spreads to the sand bed.
Read this carefully. I am not advocating adding fish such as Tangs to an aquarium until the aquarium is mature. There is a lot going on with the water in a marine system and it needs time to mature. This has nothing to do with ammonia, nitrite, or nitrate. There is something to be said for "aged" water. Mature systems "feel" different than newly set up aquariums. I suspect the development of algae and populations of copepods, amphipods, etc have a lot to do with this. Most fish have a higher survival rate in established aquariums, which I consider to be aquariums that have been set up for 6 months or longer.
Now, the original question. In freshwater systems nitrosomosa and nitrbacter bacteria process ammonia and nitrite. These are not the same bacteria that accomplish this task in saltwater. I forget the names of the saltwater variety and could easily google it, but i will leave that for you.