"They want a dog, I want a cat."
Translation:Ei doresc un câine, eu doresc o pisică.
First, in this type of construct that communicates a difference (X does something, Y does something else) we don't usually drop either of the subjects (X and Y - "ei" and "eu" in this particular phrase). They have to be present to make it clear the parts, who wants what.
Second, it happens that the conjugated forms of the first person singular ("eu") and the third person plural ("ei") are identical in the present tense: "doresc". From a sentence like "Doresc un câine, doresc o pisică." it would be impossible to say who wants what.