There are several synonyms for the phrase "hand round," which means to distribute or pass around something. Some alternatives include "circulate," "dispense," "dole out," "give out," "pass out," "share," and "offer." These terms are commonly used in various contexts, such as distributing food or drinks during a gathering, sharing documents or information at a meeting, or passing around items for inspection during a group activity. Depending on the situation, some of these synonyms may be more appropriate than others, but they all convey the idea of transferring something from one person to another in a collective manner.