As others have indicated, there are many different notions of random 3-manifold or random link. Here are two other types of models of random linking:

 1. The Petaluma Model (http://front.math.ucdavis.edu/1411.3308). This model is based on link diagrams with a single multicrossing, with the randomness given by a choice of random permutation, which determines the heights of the arcs relative to one another. They're actually able to compute the distribution of (appropriately scaled) linking numbers on the nose with this model. I don't think that this has been done with any other model.  They do some computations of some of the moments of other low order finite type invariants too.

 2. The random projection model (http://front.math.ucdavis.edu/1602.01484). This model starts with a fixed embedding of some circles in some high-dimensional Hilbert space, and randomly projects these onto a 3-dimensional subspace. In principal, the moments of the linking numbers ought to be computable. This is an intriguing model because there are continuously many parameters. It's possible that by varying the initial embeddings, these models can be made to limit to other types of models. Maybe that could explain some of the universality observed experimentally and discussed in the Petaluma paper.

As far as I'm aware, no one has examined hyperbolicity in either of these models.