The Sign of Peace (Kiss of Peace) is accomplished in the Traditional Rite with the celebrant saying ‘Pax Domini sit semper vobiscum’ (‘The peace of the Lord be with you’), as he thrice makes the Sign of the Cross with a fragment of the Host over the Chalice. The symbolism here is that the ‘peace’ comes from the Lord, after the Consecration has been accomplished.
The people then answer ‘Et cum spiritu tuo’ (‘And with your spirit’), thereby signifying their communion with the priest and with each other. The priest then extends a liturgical kiss to the deacon, who conveys it to the the subdeacon and the rest of the clergy sitting in choir.
This is in contrast to the Novus Ordo (New Mass) where there is almost no symbolism in comparison to the other Christian rites, East and West. The modus operandi in the Novus Ordo is that of a proclamation followed by a response, as if the celebration was horizontally within the community with no vertical component. (The prior statement is an allusion to Cardinal Ratzinger’s discussion on ad orientem and versus populum celebrations.)
Along this train of thought, for the Kiss of Peace to be accomplished in a proper Christian liturgical fashion, it must emanate from the Altar, and passed from person to person, just as it is done in the Traditional Roman Rite, the other Western Rites, and among the Eastern Churches.