You don't strictly need a car if you plan to stay in one town — Budva, Kotor, or Herceg Novi all have walkable centres and regular bus connections to nearby beaches. Local buses (crna gora linija) run frequently along the main coastal road and are cheap.
However, a rental car unlocks a lot more of the coast. Remote gems like Zanjic and Mirista on the Lustica Peninsula, Buljarica near Petrovac, or the long sandy Velika Plaza outside Ulcinj are difficult or time-consuming to reach by public transport. Parking near popular beaches in July and August fills up early, so plan to arrive before 9–10 am.
If you're island-hopping between several towns over a week or more, renting a car for a few days mid-trip is often the best compromise. Roads are generally good but can be narrow and winding through the Bay of Kotor and Lustica.