I've read that while the writer/screenwrite creates the story, it's the director's vision that makes the motion picture. So, whose picture is it, anyway?
In addition to other answers, that already pointed out the collaborative nature of film-making (and thus the complexity of making a definite decision "whose picture it is"), I'd like to share some additional info, be it only to further emphasize how difficult a definite answer actually is. But beware that this information is entirely Wikipedia-based and I'm far from an expert on the topic.
First of all, there is the so-called Auteur Theory, which emerged in France in the 50s (first and foremost advocated by François Truffaut and fellow critics from Cahiers du Cinéma), which said
So they basically tended to place the director as the primary creative force behind a movie and encouraged directors to actively imprint their own visions in the final product, favouring directors who acieved to mark their works with a distinct and consistent audiovisual style.
This might have been a direct reaction/aversion to the Hollywood Studio System, in which movies seemed to be an entirely industrialized process with the writer/director/cinematographer/actors simply being employees to produce the movies for the studios' profit. And this artistic and director centric approach later ushered in the movies and directors of the French New Wave (Nouvelle Vague) (and probably also the New Hollywood as a consequence of the collapsing Studio System).
Yet, this director-centric view was also frowned upon from different sides, mainly because film-making is a collaborative process and people saw that fact hurt by privileging the director that much. Special criticism came also from writers, who disagreed with the idea of directors being more authorial than screenwriters (and exactly for shifting the attention to the screenwriter, David Kipen developed the so-called Schreiber Theory as a reaction, yet in turn a rather intentionally exaggerated reaction it seems) and from people that claimed that Auteur Theory did not withstand the economic and industrial realities of film-making.
Personally speaking, I am a bit torn on the topic myself, too. While I agree that the director is usually the person with the most artistic influence on the final movie (which is so much more than just a story) and definitely deserves credit for that (in the end Auteur Theory didn't emerge out of nothing), the director-centric view of modern culture (maybe largely influenced by Auteur Theory and its decendents) sometimes tends to be exaggerated too much, in my opinion, especially when the screenwriter gets forgotten in discussions about a movie's plot. Film-making is a collaborative effort and the director is by far not the only major contributor, even if maybe the most important one.
But so much to the artisitic aspect of this whole question. With movies usually being industrial products there is however also a legal and economical aspect, which might not be any easier to answer, but nevertheless should have a more definite answer due to the more uncompromising nature of law.
Given that it is indeed usually the producers or the studio that give their money for the realization of the movie they in the end carry the financial responsibility of it and deserve to be called the "owners" of the movie in an economical sense, I guess (and are also the ones who sack the Academy Award for Best Picture for that matter).
Though, copyright law is another legal aspect that needs to be considered, and which can vary from country to country. For example Wikipedia says that under the influence of the afore mentioned Auteur Theory:
But I have a strong feeling that this whole matter of copyright laws and legal ownership is far more intricate and needs to be investigated further. But suffice for now to say that there is both an aritistic/creative aspect of ownership as well as a legal/economic one and they both might not always walk hand in hand.
All in all the whole matter of "whose picture it is" can most probably not always be answered in a clear or definite way and might even vary from movie to movie. Yet, I think modern culture seems to largely agree that the director is generally more important than the screenwriter, however correct that might be to you or to reality.
Movies are always collaborative efforts, unless the writer, director and cinematographer are all the same person:
And even if they are the same people, actors on set can always adlib which can give scenes more depth than was originally described.
The general idea is that:
Theater is the writer's medium. TV is a producer's medium. Cinema is a director's medium.
Each of these general idea's come from who is being billed as the "storyteller" of a particular piece.