Yes. Public toilets and baths would have been the most common setup, even amongst 1st Class passengers.

Referring to the deckplans (which I fully encourage everyone to browse through for fun), you can see most of the rooms in 1st Class didn't have a private bathroom.

There are three setups in 1st Class. The first (and most expensive) would be having a private bathroom, found mostly on B-Deck. I counted 30 suites total that had this feature.

The next step down would be sharing the bathroom with one other suite, found mostly on C-Deck. I count 46 rooms that would have shared a bathroom with another room. Keep in mind that these rooms could be linked together on purchase, so it was set up in such a way that a family could occupy both rooms.

The final and most common setup would be public facilities. Over 200 rooms by my count out of the permanent 1st Class rooms (rooms that would not have also been used for 2nd Class passengers).

Do feel free to check my counting. I could have easily miscounted.