I am not sure I understand the question, see the remarks of Qfwfq. Here is something related which might nevertheless be interesting for some branches of applied mathematics: an algorithm for parameterizing the intersection of two arbitrary quadrics, given in implicit form with integer coefficients. There are several research reports and even an implementation. You can even input your coefficients on-line on the web and see the result. Maybe there is something easier for the special case of two circular cones.
ADDITION: Apparently they changed the structure of their web pages; The work of these people is based on solid algebraic work, which might be what you are looking for, in their publications; then (what I find remarkable), they often go through and implement their results (see the web page about the work in their lab; I hope the links work now. in particular the work about Constant-complexity geometric problems and algebraic invariants). Those algorithms, while producing "numerical results", do so in a completely reliable way, in contrast to what you conventionally expect from numerical floating-point computation. O.k. you don't want computations; you also don't want the "sum of squares=0" solution, but what DO you want then? I suspect you are interested in the real solutions, so there will be case distinctions, 0, 1, 2 components etc. How should a potential solution to your problem look like? A single polynomial? In which variables? What should the solutions to the polynomial describe?
When you write: "The solution should describe directly what the intersection "looks like" from the point of view of the origin". Does that mean you want the projection of the intersection curve from the origin? It will be some intervals on the circle which is the projection of the whole primary cone. Do you want the endpoint of these intervals?
Another group that you might check out is the work of the EXACUS project in Saarbrücken.