I don't know exactly how to do this because each variant may have (and often has) several functions (usually corresponding to several overlapping genes, as far as I understand).

Any given variant can indeed overlap multiple genes that may, for example, share the same promoter; however, each variant can also have different functionality within the same gene's transcript isoforms. A variant can be exonic in one isoform, for example, but intronic or splicing in another.

SnpEff does output a summary file: http://snpeff.sourceforge.net/SnpEff_manual.html#outputSummary

Variant Effect Predictor also outputs a HTML stats file, which may be of use to you: