Does size matter?

It depends. Generally, for sperm competition, species where sperm from many males must compete (polyandry and polygynandry) to fertilize egg(s) have significantly larger testes than in species where sperm from multiple males does not have to compete (monogamy or polygyny).

This doesn't say anything about phallus size.

Research in rodents indicates that phallus size is also correlated with the strength of sperm competition. These results, however, do not extend to either bats or primates (humans and chimpanzees are a great counter-example).

Alternatively, most birds do not have a penis, but a cloaca (an opening through which semen, and urine pass), similar to the females, but some birds, such as ducks to have a phallus. In fact, the Argentine lake duck has one of the largest phallus to body length ratios in the known word (2:1). Why have a phallus when most birds don't need it? Because many ducks engage in forced matings. The correlation found in mice, also holds in ducks, where species with more sperm competition have larger/longer phalluses, but the really cool stuff in ducks is that they also studied the females.

The female reproductive track is not only complementary to the long, corkscrew phallus of the male (to accommodate desired matings), but there are several "off-shoots" to prevent insemination by forced matings.

Neat stuff!