Disclaimer: I am not a pastor or trained theologian.
(Other disclaimer: Since your question asked about a Protestant perspective, this answer is written from that perspective with respect to various claims made. Feel free to mentally sprinkle "according to Protestants" in front of everything.)
That said... your main question appears to be about private confession, but you also mentioned penance, so I will address both.
My guess is that requiring private confession would be seen as a form of works righteousness. "For by grace you have been saved [...] not a result of works" (Ephesians 2:8-9). Salvation is the free gift of God; there is no act we can do as humans to "merit" salvation, and so saying that anything, besides wad Christ has already accomplished, is necessary for salvation would be considered by many Protestants to be heretical. (For more information, see what your Protestant/Reformed denomination of choice has to say about "sola gratia" / "grace alone". I'll omit links because Wikipedia's article is unhelpful, and for anything else I'd likely have to cite a source from a specific denomination.)
(I won't go into the rabbit hole of whether repentance is "necessary". On the one hand, it is. On the other, we are incapable of repenting on our own, and it is only through the Spirit working within us that we can repent. However, there is no specific mechanism of repentance required.)
That said, there are means by which God's grace is made manifest... one of which is confession. At least some Protestants commend parishioners to engage in private confession, as there are significant benefits (for one, the relief of conscience to be had from receiving forgiveness one-on-one after confessing specific sins). However, it is not required.
Protestants, AFAIK (Evangelicals/Lutherans, certainly) mostly reject penance, for much the same reason; it leans very much to being a form of works righteousness. No act we can do as humans can make up for our Sin; only Christ's death can accomplish that. And since Christ already died for us, and His redemption is complete (John 19:30), no act we can do can "add" to what He has already done.
That's not to say that some form of penance (and here I'm thinking more something that an individual chooses for themself, rather than something "assigned" by a priest) might not be salutary, but it must rightly be an act which is undertaken freely, and not because it is seen as "necessary" or as having some benefit with respect to our salvation.