Is there a way to compute something in a first shader program (eg. screen-space coordinates from word-space ones) and access those results in an another shader program?
Is there a nicer solution?
Yes you can, this is how deferred rendering for example would work. For that you would render to a texture in one pass, in openGL this would be via an FBO, then feed that texture as the input to the next pass/shader.
While the other answers are giving possible techniques to achieve what you're looking for, rendering your stuff into a texture is not always the best way if you need it as VBOs (i.e. vertex attributes) in the next render step (requiring either a read into a PBO and some index/coordinate/sizing magic or a vertex texture fetch).
An approach that would more accurately represent your 2nd point and be better when you need a transformation from a buffer into a buffer would be what's called Transform Feedback (or Stream Output in the DirectX world). However, this requires OpenGL 3/DirectX 10 hardware (however, not necessarily the GL4/DX11 needed for compute shaders). It's also a little less of a synchronization mess than compute shaders with arbitrary shader storage buffers when your use case is really just to transform a bunch of vertices/primitives into a nother bunch of vertices/primitives.
It basically lets you hook into the render pipeline after the vertex (or rather primitive) processing step and before the rasterization and write out whatever per-vertex values you computed directly into a buffer object (that you can then use as vertex attributes in the next step). It therefore doesn't draw anything into the framebuffer either. You just specify a bunch of varying/out variables and how to format them into the buffer and bind a buffer to the corresponding binding point. You can also use a query to gather information of how many primtives were written (since a geometry shader could change that dynamically) or, in later versions, just render the results directly without any need for synchronization. For more details, see the above Wiki article.
Together with the geometry shader (which TF hardware should have, too) and its abilities for changing the primtive type or removing/adding vertices, this makes a useful tool for transforming your geometry however you deem fit in preparation of further rendering stages. Of course, it has to be evaluated if that really buys you anything in the particular use-case, especially if you're really just doing a simple vertex transformation.
