Two different local parishes I attend handle it without touching. My understanding is that this prevents two things: physical disruption of the litugy (the tendency to leap across aisles or otherwise strain to touch others out of some self-conscious sense of compulsion), and concern/revulsion about other people’s personal habits (indelicately referred to in some detail already here

), as well as the problem of possible contagion.
So what happens at these two (very different styled) parishes is that people look at others around them (and in other rows) and smile and speak softly, and sometimes gesture with hands (such as a slight bow with hands folded, or a slight, respectful hand gesture – no distracting flourishes). The most important thing seems to be the eye contact.
Here’s another problem that perhaps hasn’t been mentioned:
In one of these two parishes, there is a daily Mass-goer who has some severe nerve disease; it could be something even more involving than Parkinson’s. (He also may be a saint; seeing him, I can’t help but believe he must suffer physically almost all of the time; yet he manages with these challenges to attend daily Mass!) It would be an imposition and an embarrassment to this person to attempt physical contact with him, even though he looks to be the sweetest person ever. I am so humbled by his very presence. He also looks relieved that no one is forcing him to try to exercise his limited powers of mobility! And that relief and joy on his face is there when I turn to him and smile my warmest smile ever. His whole face lights up with joy.
That’s all that is needed to communicate The Peace of Christ!
Finally, I’m not sure that people have mentioned these variables:
~People who don’t wash their hands. Sorry to be indelicate, but this can be a real problem. You touch their hands; their hands are obviously dirty/sticky, and this is by no means limited to children, but can be very common in grown men, and I’ll just leave it there for delicate sensibilities.
~Men who apparently never learned to shake a woman’s hand – women who have one-third to one-fourth of the muscle strength that a man has. I do not come from frail genes, but my hand has been injured so often at these hand-shakes (and hand-holding during the Our Father) that I now refuse to shake a man’s hand during Mass unless I am acquainted with his particular grip already.
Some men take a woman’s fingers and crush them. Others seem to think that brutality is a sign of affection. Go figure.
On the constructive side, here is how I would prefer to communicate the Peace of Christ: Biblically. That means that if indeed I know someone in that congregation that I am not at peace with, I approach that person before Mass (if I see him or her) and express peace/ forgiveness/ reconciliation, etc. And if I don’t know such a person, I think it would be wonderful to select one stranger, as one is entering the nave, and wish that person peace, warmly and sincerely – again, before Mass. )Sometimes, for example, after Mass, I will indeed touch someone or someones, but not necessarily their hands. I’ll touch them on the shoulder affectionately. Maybe it’s because I also like people to do that to me.
Then, the priest, after the Our Father, summarizes for the whole congregation that Message of Peace (as the priest now already does) to all of us as one gathered body, after we have done that individually. That way would not “omit” the ritual but handle it differently.