The celebration of the Mass is regulated by a document called the General Instruction of the Roman Missal (GIRM), so in order to answer most questions about practices such as postures or positions at the Mass, the GIRM is a good place to start.
Regarding the laity’s action at the Rite of Peace, GIRM No. 82 says: “As for the sign of peace to be given, the manner is to be established by Conferences of Bishops in accordance with the culture and customs of the peoples. It is, however, appropriate that each person offer the sign of peace only to those who are nearest and in a sober manner.”
The current GIRM was promulgated in 2002, but a 2014 circular letter was sent to conferences of bishops throughout the world by the Holy See’s Congregation for the Divine Worship and Discipline of the Sacraments. In response to that, the U.S. Conference of Catholic Bishops said this, which gets most fully to the heart of your question: “No ‘official’ expression of peace has ever been stipulated for the dioceses of the United States. Perhaps the most common form for the exchange of peace in this country is shaking hands, but the diocesan bishop may encourage other forms as well for cultural or other pastoral reasons.”
Michael R. Heinlein is editor of Simply Catholic. Follow him on Twitter @HeinleinMichael.