The longer answer looks at many different factors that may influence who does the asking, and how he or she does so.
So, who does do the asking in a same-sex couple? The short answer is: whoever wants to. In same-sex couples it can be more difficult to figure out. For example, in opposite-sex couples (which may include people who are bisexual or transgender), traditionally the man is the one who proposes.
Now that marriage equality is the law of the land, some LGBTQ couples who are planning to get married are forced to figure out some of the logistics that were traditionally defined by gender.