Yes, depending on one's definition of "Starship".
I think that it is possible for private Federation citizens to legally more or less own or possess space ships that fit (1) but not (2):
- 1) Common and usual definition: A space ship that travels interstellar distances, usually with a FTL drive.
- 2) TOS-era definition: A top of the line Starfleet space ship. Starships seem to be multi-purpose vessels as powerful as the top military vessels in Starfleet and with all the labs of a top science vessel. Starships are used for exploration, patrolling, diplomacy, emergency response, scientific research, and running routine errands.
This exchange in "Bread and Circuses" delineates between the the two.
KIRK: I've had to select men to die before so that others could be saved.
CLAUDIUS: You're a clever liar, Captain Kirk. Merikus was a spaceship captain. I've observed him thoroughly. Your species has no such strength.
MERIK: He commands not just a spaceship, Proconsul, but a Starship. A very special vessel and crew. I tried for such a command.
Merik clearly makes a distinction between an interstellar spaceship - a starship by the first definition - and a Starfleet starship.
Further, in the era of TNG another definition which seems to be stricter than (1) and looser than (2) exists (at least based on fan discussions which assume that every class of Starfleet ship is a starship) - The research vessel Vico in "Hero Worship" may have been such a Starfleet vessel.
Another example from TNG: The Odin in "Angel One" was a freighter and the crew claimed that as civilians the Prime Directive didn't apply to them.
RAMSEY: Mistress Beata be damned! Her wish is not my command, and neither is yours. You can't force us to go.
DATA: Mister Ramsey is correct, Counsellor. The Odin was not a Starship, which means her crew is not bound by the Prime Directive. If he and the others wish to stay here, there is absolutely nothing we can do about it.
So either the Odin was a private freighter with a civilian crew not bound by the Prime Directive or it was a Starfleet freighter with a Starfleet crew not bound by the Prime Directive. In the first case it proves there are private interstellar ships in the era of TNG, in the second case it proves that not all Starfleet interstellar ships are Starships with crews bound by the Prime Directive.
It is my belief that an officer could have the rank of captain and the position of spaceship command, but that "Starship command" was a special subcategory of ship command in Starfleet.
The biography of Kirk in The Making of Star Trek says (paraphrased):
"Kirk was the youngest graduate of Starfleet Academy to be assigned as a starship command captain."
That strongly implies that a Starfleet officer could be in command of a different type of ship not classified as a Starship.
IN "Court Martial" the courtroom computer says:
COMPUTER: James T. Kirk, serial number SC937-0176CEC. Service rank, Captain. Position, Starship command. Current assignment, USS Enterprise. Commendations, Palm Leaf Of Axanar Peace Mission, Grankite Order of Tactics, Class of Excellence, Prantares Ribbon of Commendation, Classes first and second
This shows a difference between having the rank of captain and the position of starship command. Obvious a Starfleet officer could have the rank of captain without commanding a ship if he was for example a high ranking supply officer or medical officer, etc.
Earlier, in a meeting with Commodore Stone:
STONE: Stop recording. Now, look, Jim. Not one man in a million could do what you and I have done. Command a starship. A hundred decisions a day, hundreds of lives staked on you making every one of them right. You're played out, Jim. Exhausted.
If Stone is literally correct the Federation must have a population of over a million "men" (however defined) for each starship captain. And that strongly implies that a Starship is a special type of Starfleet vessel because in real life being able to captain an ordinary type of sea ship is far more common than one in a million.
Note that officers from the survey vessel Antares in "Charlie X" wore Starfleet uniforms and yet the ship had only 20 men aboard.
KIRK: A survey ship with twenty men aboard lost. No reason. Obviously, Captain Ramart was not aware of any trouble. I can't figure it.
Twenty men is a lot less than the hundreds that Commodore Stone said were in the crew of a starship. Thus it seems likely that Starfleet has many small ships with small crews that are not Starships.