I guess "their" should refer to one of the nouns, that is, either "Your friend" (more likely) or "you" (not so likely but possible). So instead of "their" it should read "his", "her" or "your".
But what if "Your friend" is married, for example, and although he is going to travel alone, the "cat or dog" belongs to the couple? In this case "their" would refer to a previous information in the text (the couple's cat or dog).
Not a native speaker
Student or Learner