While Beatrice and Benedick fight with words throughout the entire play, they are seen as the classic "dueling lovers," or a romantic pairing that is exemplified with good-natured, if sometimes harsh, bickering. Every time they meet, they exchange insults and attempt to best the other with words and puns.
In terms of having power in the relationship, both Beatrice and Benedick are specifically and vocally opposed to marriage, but they harbor feelings for each other, and neither wants to be the first to admit it. In fact, during the play they are portrayed as being all but equal both in influence and wit; it is rare for either to gain the upper hand on the other. However, until the scene in which all leave and they confess their love, her barbs are somewhat more fierce than Benedick's: consider these lines from the masked dance.
BEATRICE: Why, he is the prince's jester, a very dull fool. Only his gift is in devising impossible slanders. None but libertines delight in him; and the commendation is not in his wit, but in his villainy; for he both pleases men and angers them, and then they laugh at him and beat him. I am sure he is in the fleet. I would he had boarded me.
BENEDICK: When I know the gentleman, I'll tell him what you say.
She insults his wit and his ability to entertain others, and he still says nothing in return, which allows her both the victory in wordplay and the satisfaction of seeing him unwilling to attack her in the public venue of the masked dance. At the same time, his gentlemanly behavior gives her some shame, as he is allowing her to attack without reprisal, and she is certain he knows who she is; if he did not, he would have said something in return, safe in anonymity.
Therefore, the power in the relationship might be said to waver depending on who has the upper hand in a specified verbal battle. However, later in the play, Beatrice convinces Benedick to kill Claudio for her after he publicly scorns Hero at their wedding. Benedick agrees, and so the balance of power falls firmly on Beatrice's side; while Benedick is still reeling from their mutual confession of love, the act of murder is quite significant, and shows how highly he regards her. While he does not actually end up killing Claudio, the fact that he was willing to based almost entirely on Beatrice's word shows that after their mutual confession, he is willing to let her be the deciding voice in the relationship.