In terms of canon law, the desecration of the sacred species is a crime, punishable by automatic excommunication. In most cases, the person does not know what they are doing or how wrong it is, so I’ll assume that is the case (rather than deliberate desecration, perhaps for satanic purposes or as an atheistic stunt.) But even if the person does not know the seriousness of what they are doing, it is still objectively a very grave matter. So we can’t just let it happen.
A pastoral complication is that people don’t like to be “shown up” or embarrassed in front of the congregation so they can easily become annoyed or aggressive in their confusion. Therefore we need to keep calm and speak in a kindly way. I would usually stop and look clearly in the person’s direction. If there is a trusted member of the congregation, I would nod to them to indicate that I want the person’s attention. As a final resort, I would leave the sanctuary and go to the person to ask them to return the sacred host to me or consume it reverently. In any case, I would briefly explain that we believe that this is the sacred body and blood of Our Lord.
In Churches where there is a greater danger of such desecration, there might be people routinely on duty to whom the priest can turn for assistance in such cases. They would need to have some instruction about the fact that this will happen usually out of ignorance and not to appear aggressive.
Now what can we do to try to prevent these incidents? First of all, we need, as a matter of routine, to give monitions at some point during Masses such as those for weddings, funerals, first Communion. Here is what I say:
"Just a note about Holy Communion: in the Catholic Church, it is practising Catholics who receive Holy Communion (people who go to Mass every week.) There are always plenty of people who don’t go to communion so there is no need to feel awkward. There is a hymn that you can join in with or just listen to while you say your own prayers."In England, that covers it more or less – the “come up for a blessing and cross your arms” thing is not really necessary any more since we are now into the generation of people so uncatechised that they do not know the “Our Father.”
As a more proximate measure, if people that I do not recognise come up for Holy Communion and extend their hands, I do not move on to the next person until they have consumed the sacred host. This helps to avoid the need for calling them back or going down the Church after them. It is much easier where the priest moves along the altar rail to give Communion: the practice of people coming up in a queue and then stepping to one side to consume the host is almost inviting the problem.
As a more remote measure, I pray that one day the general practice returns whereby everyone kneels for Holy Communion and receives on the tongue. (See Communion kneeling, on the tongue, for more about this.) This would not prevent the determined Satanist or atheist from their malicious desecration but would prevent the ignorant from walking away with the sacred host blithely unaware of the desecration they are committing. This is not a purely personal concern. In Memoriale Domini, Pope Paul VI said:
"Further, the practice which must be considered traditional ensures, more effectively, that holy communion is distributed with the proper respect, decorum and dignity. It removes the danger of profanation of the sacred species, in which 'in a unique way, Christ, God and man, is present whole and entire, substantially and continually.' Lastly, it ensures that diligent carefulness about the fragments of consecrated bread which the Church has always recommended"(For more on Memoriale Domini, see Memoriale Domini - a reminder.)