A short greeting before Communion, not an intermission.
Offer it to those immediately around you. A handshake, a nod, or a word is enough.
Keep it brief. The priest does not leave the sanctuary for it, and neither should anyone else.
It comes before Communion for a reason. If you are at odds with the person next to you, that is the moment to mean it.
You cannot receive the body of Christ while refusing the body of Christ standing beside you. The gesture makes reconciliation a condition of Communion rather than a private feeling.
The kiss of peace is apostolic. Saint Paul closes several letters by telling the churches to greet one another with a holy kiss, and the early liturgies placed it near the offering.
In the Roman rite it drifted to just before Communion, tied to the Lord's words about leaving your gift at the altar and being reconciled with your brother first.
The Church has repeatedly asked that it stay sober and short, precisely because it means something.
The rite may be omitted entirely at the priest's discretion, and no one is obliged to shake a hand. Illness is a perfectly good reason to nod instead.