In his book Yofi Shel Ivrit (chapter 31), Avshalom Kor points out a distinction I had never really paid attention to before: the difference between rashut רָשׁוּת and reshut רְשׁוּת.
Reshut (with a shva) means "permission". It comes from the root רשה meaning "to authorize, permit, allow". It is related to the word rishayon רשיון - "license". Reshut hadibur רשות הדיבור - "permission to speak".
Rashut (with a kamatz) means "authority". For example, the Postal Authority is rashut hadoar רשות הדואר, and the Broadcasting Authority is rashut hashidur רשות השידור. Kor writes that rashut comes from the word rosh ראש - "head". Klein agrees, and writes that the word is "probably a shortened spelling for rashut with an alef - ראשות". Today rashut with an alef means "chairmanship" and rashut without an alef means "authority". Kor gives the following example:
הרשות המחוקקת ברשאותו של ברק בן אבינועם
"The legislative authority (rashut without an alef) is under the chairmanship (rashut with an alef) of Barak ben Avinoam".
However, Klein also adds that:
Several scholars and lexicographers derive rashut רשות from base רשה ( = to empower, authorize).
But Kor writes that there are some sources that use the word ראשות (with an alef) instead of רשות rashut for "authority" - so I'm not sure which etymology is correct. Ben Yehuda writes that the meanings of the two words - reshut and rashut - are so similar, and the sounds are very close that it is impossible to seperate them (and therefore determine the precise etymology.)
Whether rashut and reshut are related or not, there certainly are cases where it is not clear which word is intended (particularly in non-vowelized texts.) For example, in the Mishna Moed Katan 3:3, we find the term אגרות של רשות. Kehati points out that the commentaries disagree whether it should be read igrot shel rashut or igrot shel reshut. Similarly, in the Talmud, Eruvin 41b, the term והרשות appears. Rabbeinu Chananel says it means "authority", but Rashi says it might have other explanations, and it is difficult to say.
What about רשות היחיד and רשות הציבור - private domain and public domain? Should it be rashut hayachid and rashut hatzibur OR reshut hayachid and reshut hatzibur? Kor explains that the correct form here is reshut. In the private domain - only the individual has permission to act, whereas in the public domain the entire public can do as they please.
One more "rashut" word that Kor doesn't mention is rashut רשות - "poverty", coming from rash רש - "beggar".