Yes — Montenegrin law requires that a public access zone is preserved on every beach, so you can always bring your own towel and sit for free somewhere along the shore. The free stretches are typically at the ends or edges of a beach, away from the organised sunbed rows.
On long beaches the free zone can be generous. Velika Plaza near Ulcinj, at roughly 12–13 kilometres, has vast stretches you can settle on without paying a thing. Similarly, Buljarica near Petrovac is mostly undeveloped. On compact town beaches like Mala Plaza in Ulcinj or Mogren in Budva the free strip is narrower and more crowded.
If you want shade without renting, a few coves in the Bay of Kotor — particularly around Lustica — have natural shade from trees or rocks. Arriving with a beach umbrella from home or your accommodation is also a common money-saving approach.