Both Newtonian gravity[1] and general relativity can be expressed in a geometric form such that the trajectory of a test particle is a geodesic. When the theory is expressed in this form, the question becomes whether or not we can recover detailed information about an entire semi-Riemannian space from knowledge of a single geodesic. Typically there is absolutely zero information contained in a geodesic; the information content comes from incidence relations between different geodesics. (You could have information about whether a geodesic is self-intersecting, but even in a spacetime with closed timelike curves we don't expect self-intersection for a geodesic in general position.)
The question really presupposes some additional machinery, such as a global Cartesian coordinate system for space plus a Newton-style absolute time. The existence of geometrical formulations of Newtonian gravity shows that this machinery isn't fundamental to the theory. From the point of view of Isaac Newton, we assume an omniscient observer O who can see the state of all matter in the universe. O can tell the difference between an object that is moving inertially and one that is accelerating uniformly because of a sheet of mass a billion light years away. Such an inertially moving object implies an inertial frame of reference K. Once we have K, we can observe the motion of a test particle and say whether it's straight or curved.
So the answer to the question is that the trajectory of a single test particle tells us nothing at all about gravitational sources. Only an omniscient O who already knows about all distant sources can even construct K and describe the trajectory as something other than a geodesic.
To extract information about the spacetime, we need incidence relations between geodesics. As an example, suppose that two test particles cross paths, and that their world-lines intersect not just at that event but at some other, later event as well. The two geodesics form a lune. This tells us something about the Ricci curvature, which implies that there is at least one gravitational source whose attraction brought the particles back together.
An example that may be helpful is a Newtonian spacetime with countably many identical masses distributed on an infinite cubical lattice, all initially at rest relative to their neighbors. By symmetry, their world-lines will never intersect. No information is recoverable from observing one such mass. We can replace one of the masses with a test particle of negligible mass, and the test particle will still never accelerate according to an omniscient O who considers the lattice's initial state to have defined an inertial coordinate system.
[1] Giulini, "Some remarks on the notions of general covariance and background independence," http://arxiv.org/abs/gr-qc/0603087