The rest of the Christian world believes that the commission at the end of Matthew applies to the whole world. This would mean that anyone (men, women, children, believers, non-believers) can baptize anyone.
Both are wrong.
Priesthood is not a permission slip. God does not give his power to men to use independent of his will. There is no such thing as freelancing when it comes to God's power. It can only ever be used exactly as he would use it, and this requires revelation from him in every case.
Because of this, it is better to focus on God's will instead of God's power. The latter is never found outside the former, and many times the former is found outside the latter. In other words, there are many cases when God's will is to withhold, not to exercise his power.
The question, "do I have the authority to baptize?" is actually the wrong question. We really need to be careful about how we structure our prayers to God and how we frame our expectations of answers. In most cases where the person does have authority, the Lord can't answer the question as phrased. The answer "no" would be incorrect, and the answer "yes" would convey more than intended, as priesthood revealed to an individual from the voice of God conveys much more than the authority to baptize.
The correct question is "under what circumstances is a person authorized to baptize?" The answer is whenever a person is operating within the will of God. In the case of many miraculous events, such as healing the sick, raising the dead, moving mountains, turning away armies, quenching the violence of fire, etc., this is a case-by-case question that requires a connection to God to answer. In the case of baptism, the Lord has revealed his will, and it is generally applicable. In other words, any case where the circumstances are fulfilled lies within God's will, and therefore commands God's authority.
The circumstances where a person can baptize are that they:
- Must have repented from all their sins.
- Must be able to ascertain whether those they baptize have repented of all their sins.
- Must know the proper form of baptism.
If you fulfill these qualifications, you are qualified to baptize. If you do not, you are not.
As you can see, these items can be transmitted from one person to another independent of the action of heaven. Rather than be dismayed or shocked by that, we should realize that God has very plainly explained these things, and made it plain that he wants everyone to come to him and be baptized. It is all meant to be very available.
But what about keys? The LDS claim that they alone can perform authorized baptism because John the Baptist came to Joseph Smith and conveyed authority to Joseph Smith, who transmitted it to others through modern times. According to the account, this authority included the ministry of angels. How many LDS Aaronic priests do you know (or any other LDS person, for that matter) who enjoys the visible ministration of angels? If laying on of hands was the effectual transmission of authority, those claiming it would also have the ministration of angels. Since they do not, why should anyone believe they have the authority to baptize by that same laying on of hands?
Does this mean that LDS baptisms are ineffectual? No. It means that if they are effectual, it is by some other means than authority received by the laying on of hands. If an LDS person, or any other Christian, fulfills the qualifications listed above, their baptism is valid.
As I describe in "Seek Ye This Jesus," the common understanding about keys as a permission slip to exercise God's power is not correct. Instead, keys are to know God's will pertaining to a situation. If you know God's will, you have the power to participate in its fulfillment. If you don't, you are just guessing, and you cannot exercise faith. It is faith that works God's miracles, not guessing.
The keys of the Aaronic priesthood are not received via an ordinance, such as the laying on of hands, though there's nothing wrong with the ordinance. Instead, they are received via instruction. Once the doctrine of repentance is understood, and the form of baptism is understood, one possesses the keys of baptism. The same goes for blessing bread and wine after the pattern the Lord revealed in the last supper. The same goes for receiving the administration of angels.
The keys of the Holy Order after the Son of God are not received via an ordinance, per se, either. However, the knowledge they consist of is acquired during that ordinance, which involves the Lord, among other things, telling a person that they have power to do the things enumerated in JST Gen 14. If the Lord tells you that you have authority from heaven to do something, it would be easy to claim at least part of the description of JST Gen 14, which is a very different thing altogether than the Aaronic priesthood.