Are non-Catholics allowed to go to confession?

Non-Catholics, being spiritually dead, are also permitted to enter the confessional, but they are obligated to reveal that they are not Catholics. There is no question of receiving Sacramental absolution: that is possible only for members of the Catholic Church. Entry to the Church involves various rites, including formal abjurations of heresies. Of course, a traditional priest would quickly realize that he was dealing with a non-Catholic. Now, I am talking about a traditional priest. A modernist Novus Ordo would probably be perfectly happy giving absolution to just about anybody, if they even use that ritual. The real answer is that non-Catholics are not permitted to go to Confession; they must first enter the Church through Baptism.

My Catholic friend answers this question thus:

A non Catholic can enter the confessional to talk to the priest in private. However, to infer in any way that it is a "confession" is sacriliege to the sacrament.

Basic theology, matter form and intention. Being a believing Catholic is a requirement for the sacrament, nothing wrong with having a chat with a potential convert.

The priest could make an appointment with the non-Catholic to chat in private with him about any concerns he might have. The point is for the priest not to drive him away, but make the non-Catholic understand that this sacrament is instituted by Christ for those who have the faith already and are baptized Catholics.

The short answer is, "Yes, he can." At a minimum, the confession of those of us came into the Church, already having been baptized, it is necessary and necessarily efficacious.

My RCIA director (a canon lawyer) basically explained that for all practical purposes, a priest has the authority to allow confession and communion to anyone, so long as it is pastorally appropriate. I have a couple of friends who went to confession repeatedly and well before they came into the Church (one left RCIA after first confession but before reception, another went to confession inadvertently and only in doing so realized the power in it (he came into the Church with me)).

Andrew's got the canon law side of things down, but I think a majority of non-Catholic confessions are said by those coming in to the Catholic Church according to the norms of the rite of Christian initiation for adults chapter 5 concerning the reception of baptized Christians (also done for Children in a separate chapter).

Any baptized Christian willing to make a profession can (and probably should) make a general confession of sins to any priest prior to reception into the Catholic Church. The confessor should be told of the intention to be received into the Church and will most likely help out with the confession. If extra time is needed, it's a good idea to schedule these by appointment rather than to make the old ladies wait before Mass (lest they have to confess more impatience than they were previously disposed to confess)

(Note: this answer references the RCIA as approved by the North American Bishops, so compulsory pre-entrance confession may exist elsewhere, as it apparently does for those entering the Church en masse from the Anglican Church to Our Lady of Walshingham Ordinariate)

Yes, a non Catholic can confess to a Catholic priest. (The canon law point has already been addressed).

What the non Catholic is not generally eligible for is the full sacrament of Penance and Reconciliation.; thus the priest cannot grant you absolution.¹ Our priest makes that clear to folks who are married to Catholics but who aren't of that denomination yet. It is OK to unburden your heart, but the full sacrament is only for those in communion with the Church.

Where this gets flexed a bit is during the Candidate process during the RCIA. A few weeks before being confirmed and received into the Church, the normal flow is that shortly before Easter Vigil, or other Confirmation event, a priest hears the Candidates' confession. (A Candidate is someone who received a valid Baptism from another denomination of Christian).

In the case of the non-baptized (Catechumens) entering the church, I've seen it done both ways.

1. The priest hears the Catechumens' first confession before baptism and confirmation.

2. Rely on Baptism washing away all stain of sin (which it does).

   After the latter(2) way of doing this, the new Catholic is eligible for and encouraged to receive (due to any loss of Baptismal grace via mortal sin, and at least annually per the "Easter duty") the Sacrament of Penance and Reconciliation (Confession).

   CCC 1446 Christ instituted the sacrament of Penance for all sinful members of his Church: above all for those who, since Baptism, have fallen into grave sin, and have thus lost their baptismal grace and wounded ecclesial communion. It is to them that the sacrament of Penance offers a new possibility to convert and to recover the grace of justification. The Fathers of the Church present this sacrament as "the second plank [of salvation] after the shipwreck which is the loss of grace."

   The Vicar General in our diocese advocates for the latter form(2), but the priests in a given parish are allowed to do either. (Ours did the former(1) for years). As I understand it, the Canon Law that Andrew cites offers that flexibility.

The above applies to those entering the church; it is not technically applicable to your case unless you intend to leave the Episcopalian Church for the Roman Catholic Church.

Any Priest whom you approach with the request that he hear your confession will be able to give you (at the very least) pastoral guidance and assistance. If you feel it would be beneficial to you, all you can do is find a local Catholic priest and ask if he'll hear your confession.

¹ Exception to the general rule: a non-Catholic Christian who is at the point of death (or in danger of death, per and makes a profession of faith to a priest, may also receive absolution via this sacrament. This is one of those edge cases. (844)

This answer was originally in this question, and was moved here.

To capture a point from two comments, and thus preserve them:

Not exactly “at the point of death;” but “in danger of death.” For example, I would gladly hear the confession of a non-Catholic soldier who was about to go on a dangerous mission, or a non-Catholic who was in an area in which a dangerous epidemic had begun, and so forth (provided he professes the Catholic faith in the sacrament, obviously). In other words, the person does not have to be actually dying to be in “danger of death.”
In other words, the impending death of the person need not be certain, or even probable—just reasonably possible. (This applies to all similar cases; for instance, in danger of death, a Catholic may go to confession to an Orthodox priest, or even a laicized Catholic priest.)

As ever, many thanks to @AthanasiusOfAlex

I am a priest who deals mostly with children. At schools, camps, and youth events we have penance services or just plain opportunities for confession. Quite a few non-catholics often come to confession. First, it is unlikely that any of the young people who come have done anything serious. It is mostly the usual childhood faults. If it becomes obvious that they are not Catholics, I ask them if they are Baptized. If they are, they are capable of receiving the sacrament, have asked for it, so I give it. If they say they aren't, I give them a blessing and assure them that God forgives all their sins. If they don't know, I might get enough information from them to indicate that they are or are not baptized and give them absolution or a blessing according to my best judgement, realizing that I don't want to withold sacramental absolution, if it is valid, but that God always forgives when a person says "I'm sorry."