I don’t condone self-baptism in any way, and such practices seem strange to me.
I don’t see indeed what authority could legitimate it, including from a Reformed perspective. I don’t know either of any church which actually practices self-baptism; I view those cases as historical oddities.
this is just another way the non-Catholic christians feel they have a right to self-interpret scripture
I have to point out, though, that self-baptism probably happened from time to time throughout Church history, including well before the Protestant Reformation.
There is for example an apocryphal text called the
Acts of Paul and Thecla, which was already extant in 190. We know that because that’s about when Tertullian wrote a scathing evaluation of the narrative in
On Baptism (interestingly enough, what bothers Tertullian is not as much the fact that the text seems to promote self-baptism as the fact it seems to promote baptism by women). The text narrates how Thecla, condemned to be martyred, self-baptises in the arena by jumping in a water tank containing ferocious seals, who are struck by lightning and die before they could harm her.
Although this episode is certainly legendary (Thecla was removed from the calendar of saints by the Catholic Church for lack of historical evidence, but she still has a feast day in Eastern churches), it still shows that self-baptism was not a completely unheard-of concept in the early Church.
Absolutely no authority at all, except for themselves.
To be fair, Smyth himself came to recognize that and repudiated his attempted self-(re)baptism in later life.