What stops the average joe from Baptizing someone?
Nothing. Baptism administered with the right intention and formula is valid. Even an atheist can do it.
Here however is the normal procedure, from Canon Law:
Can. 861 §1 The ordinary minister of baptism is a Bishop, a priest or a deacon, without prejudice to the provision of can. 530, n. 1. [This canon speaks of the privileges of parish priests.)
§2 If the ordinary minister is absent or impeded, a catechist or some other person deputed to this office by the local Ordinary, may lawfully confer baptism; indeed, in a case of necessity, any person who has the requisite intention may do so. Pastors of souls, especially parish priests, are to be diligent in ensuring that Christ’s faithful are taught the correct way to baptise.
Can. 862 Except in a case of necessity, it is unlawful for anyone without due permission to confer baptism outside his own territory, not even upon his own subjects.
Can. 863 The baptism of adults, at least of those who have completed their fourteenth year, is to be referred to the Bishop, so that he himself may confer it if he judges this appropriate.
Can. 868 §1 For an infant to be baptised lawfully it is required:
1° that the parents, or at least one of them, or the person who lawfully holds their place, give their consent;
2° that there be a well-founded hope that the child will be brought up in the catholic religion. If such hope is truly lacking, the baptism is, in accordance with the provisions of particular law, to be deferred and the parents advised of the reason for this.
§2 An infant of catholic parents, indeed even of non-catholic parents, may in danger of death be baptised even if the parents are opposed to it.
Baptizing a child without parents’ consent is wrong, a very serious matter.