This is such an interesting question!
It probably depends on the gene, but in general we've found a lot of dominant alleles to be gain of function, whereas recessive alleles tend to be loss of function. When this is the case, usually one gain of function is enough to show the dominant phenotype, but both alleles need to have a loss of function to see the recessive phenotype.
Take for instance, tumor suppressor genes and proto-oncogenes. Recall that the function of proto-oncogenes is to promote cell growth. When we have a mutation, we have unregulated cell growth, aka a gain of function. We call oncogenes a dominant mutation because you only need one copy of the mutant allele to have unregulated cell growth. Even if one allele is normal, the mutant allele is enough to cause the unregulated cell growth phenotype.
On the other hand, tumor suppressor genes inhibit cell growth and check for DNA damage. When we have a mutation in a tumor suppressor gene, we have a gene that no longer checks for damage, aka a loss of function. However, even with one bad copy, the cell cycle can be kept in check because the one normal allele is able to do the job irregardless of the mutant allele. Therefore, you won't see the phenotype. You would need both mutant copies to see the phenotype of damaged cells progressing through the cell cycle, so we call this allele recessive.
Hope this helps!