Since I'm doing GO annotation just now, I came up with the same question and landed here. Here's my 2p, without too much thinking...
In favour of keeping up- and down-regulated genes together: other things being equal, you should have more power because you rely on larger sets of DE genes. Also, you test fewer hypotheses so fewer chances of false-positive calls.
In favour of separate analysis: imaging you have 5 genes up-regulated and they are all in the same pathway; you also have 50 genes downregulated scattered across pathways. I think that testing the 55 genes together will obscure the pathway enriched for the 5 genes. Is this a far-fetched scenario? Dunno, but I don't think so... You can have a treatment that specifically affects a pathway and many genes move in the same direction in that pathway (e.g., genes associated with DNA replication). This in turn causes havoc in a number of other pathways in a seemingly random manner.
I think I'm leaning towards separate analyses but I can easily be persuaded otherwise. In practice, try both and try to pick the most sensible results without fooling yourself.
Yes, I have seen some papers where upregulated and downregulated genes are categorized separately on GO term;
Once you get significant level of GO term in the subset of genes, you can go for pathway or network analysis to see the enrichment of your detected genes in certain pathway or process
C: Hypergeometric test in GSEA