In the movie Titanic, when Jack first sees Rose for the first time, there are some dogs being taken for a walk on that deck by an officer. But when the Titanic is going down, none of the dogs were ever rescued on the boats or even shown drowning. So what exactly happened to them? Were they tied up, and as a result drowned with the Titanic?
In the film (as far as I know) there are no indications that any dogs were saved and it can probably be assumed that they "went down with the ship" and were left behind with their owners. It was probably deemed by the filmmakers to be unimportant compared with the other plots being developed.
However in reality according to Premier Exhibitions
Out of the twelve dogs onboard only three survived Titanic’s tragic sinking, a Pekinese and two Pomeranians. The Pomeranians were brought onto lifeboats by their mistreses, Margaret Hays and Mrs Elizabeth Barrett Rothschild. A Pekinese named Sun Yat Sen was saved by his master, Henry S. Harper, in Lifeboat 3.
I think it's safe to assume these were fairly wealthy passengers and the ethics of saving dogs versus people was not discussed in great detail at the height of the crisis.
Did the dogs have any specific role in the story? Why do you think it was required to show the fate of some random dogs on a sinking ship? No one knows what happened to them. Probably they were still taken by their rich and spoiled owners.
Remember that scene from the movie 2012, where rich folks booked a place in the ship for their dogs while there were so many human lives at risk.
That is one possibility, otherwise they were abandoned by their owners who were running to save their own dear lives.