Who really owns the pocket watch, Richard or Elise?

Neither of them and both of them.

This is a sort of "chicken and egg" situation, or in Sci-Fi parlance, a paradox, specifically, in this case, a "Booststrap paradox".

The bootstrap paradox, or ontological paradox, is a paradox of time travel that refers to scenarios whereby items or information are passed from the future to the past, which in turn become the same items or information that are subsequently passed from the past to the future - this creates a circularity of cause-effect such that the items or information have no discernible origin. Thus, the paradox raises the ontological questions of where, when and by whom the items were created or the information derived.

After information or an object is sent back in time, it is recovered in the present and becomes the very object or information that was initially brought back in time in the first place. Numerous science fiction stories are based on this paradox, which has also been the subject of serious physics articles.

This example actually appears farther down the page under Physical Items:

An old woman gives a young man a watch; the young man then goes back in time and hands the watch to a young woman; she later grows into the older woman who hands the watch to him. The watch therefore has no point of origin.

It goes on to say something that you might not think about... if there is such a paradox, shouldn't the watch itself age and eventually stop working?:

A further paradox present for any physical item is that the watch should age each time around the loop and eventually wear out. Bringing back a copy of the watch would prevent this "wearing out" issue as would it being a "grandfather's axe".

Such an interesting question: let's ponder for a moment... Richard is an intense soul who loses his passion for the theater and love almost a decade after his graduation from college. In a sense, he is actively searching for something, not certain what. However, he ponders subconsciously seeking a focal point to which he can connect his "apathies." Lest remember the title of one of his plays: Passionate Apathies.

He keeps with him the watch given to him by the mysterious elderly woman a decade earlier that invited him, "Come back to me." In the interceding time, however, the watch has not been his true focus. It is actually not clear to us that he still possesses it until he is dining in The Grand Hotel's restaurant before visiting the portrait gallery.

When he first sees Elise's portrait, he is captivated by her almost ethereal beauty and a subtle, playful invitation in her gaze. It is because he finds nothing to connect to in the concrete world that he spends an effusive amount of time solving the mystery of the woman in the portrait.

Through research and via his own active imagination, he wills himself forward in time; yet, he is unable to do so without leaning on the advice of those who have preempted him in this endeavor, namely a former college professor. As he tells Richard, "You must remove anything that reminds you of the present or those surroundings to transport your mind to the other time and place."

Foreseeable, it is, that he remains in a rut. In fact, when he wakes up in the past, the furnishings boast similar patterns to the present and it takes a moment for the shimmering fade in/fade out to complete before he is fully invested in the scenario.

Much like scenes in a play that are rewritten, rejected, and revised, the character of Robinson validates Richard's own emotional turmoil and apathy. Elise is the "Spring" of his best-selling play "No More Spring." She represents a rebirth, revitalization of spirit. The idea that age and time are both measured but not truly marked is implied throughout the film. We remember people as we desire to do, not always as it was. In Richard's case, this causes an extreme chasm with his current reality. Was she truly inviting him into the past, or perhaps, this was a reference to their souls being reunited in Heaven?

The idea that artist such as Manet, Monet, Van Gogh, Serruat, and the like went mad for their art or sacrificed sanity to die unto themselves for the cause of it, can be duly witnessed in the subtle, gentle tones of muted cinematography. We see several scenes, especially those on the lawn and along the shore, that seem to be inspired by those paintings.

The watch is a symbol and can be assumed to be either concrete, metaphorical, or both; however, as to the initial question: who owns the watch? No one truly owns time, it is simply borrowed or on loan if you go with a metaphorical interpretation. If you choose a metaphysical course, though, it may be assumed to be Elise's who gifted Richard with the watch as he reminded her of someone she'd once known. How then can we account for similar/same names. Did she go their purposely,or was the allusion to no more spring really a commentary on time: springing forward and back as we often do in memory, reflection, dreams, and imbibed with imagination which knows no bounds?

When Richard unknowingly fingers a coin in his pocket after he and Elise make love, it seems on a surface level to be what leads to the chasm with his current reality closing. However, were it not the coin but actually the watch that separated them, then no subplot or backstory would exist. Remember, he purchased 1912 coins before hypnotizing himself once again and heeding the advice of the professor from earlier; yet somehow, their was a lone coin with a date of 1979 or 1980. In of itself, those are just numbers, yet his subconscious reminds him this is not the case.

The watch is all but forgotten as he focuses his attention on a new object but one that elicits delusions of madness, depression, and morbidly death. When Arthur and the house doctor locate him in his catatonic state and lament on its senselessness, we see this as another window to Richard's mind. He then chooses to focus on the goodness of two men who don't even "know" him and the light that brings to him leads him back to Elise as it has restored his faith in spring eternal.

We may pick up objects and lay them down,casting them aside several times a day without any really thought to their significance. Money brings him no joy and time cannot be bought or sold. It is priceless. The watch, therefore, is a link but it is what is done with it or outside of it that impacts it's true power. Elise never lost hope and thus she was able in this detailed dream to manufacture the perfect ending to a star-crossed love affair. If she could have gone back, she would have done it differently, but she imagines him crossing time and distance to validate her own love and now we know what the dialog symbolizes in the play and upon the first meeting when she inquires: "Is it you?"

She doesn't know what true love looks like or doesn't see it clearly in the face of Robinson,so she concocts different circumstances and time periods to answer her own uncertain fears. A play within a play: let everyone examine their own conscience. Do we wait and do nothing or act on instinct without forethought to the consequences. Thus, the question, the critic asked at the beginning of the film has been answered. It is free will, our choice, and what we do or don't do can yield various possible outcomes. Thus conscience and love and their paradoxes mark seconds as we wait to determine what we will do this time.

A play within a play. Far-fetched perhaps, but that is what I believe.