A public activity is one that happens in the presence of, and is witnessed by, others. A semi-public act is one in a private setting, where a controlled and limited audience can watch.
Public vs. private chastisement
When it comes to punishment, especially corporal punishment, throughout history there have been strong competing opinions both in favor of executing the same in public and in private. A public chastisement was thought to be more effective as it was more embarrassing and shameful, and it had the asset of a deterring effect on the audience. The public execution also could ensure that the chastisement was given in accordance to the law and no less or more severe than it should have been. In some religious orders, flagellation was inflicted in the presence of all inmates, in accordance with the injunction of the apostle, "Confute sinners in the presence of all".
On the other hand, it was seen as inappropriate to subject the delinquent to the shame of a public chastisement. This philosophy coined the idea "praise in public, chastise in private". The flagellation of criminal offenders which traditionally took place in the streets (bound to a cart) or on a market-place, e.g. on a whipping post, such that as many people as possible could witness it, was gradually transferred to indoors and private places. School corporal punishment would no longer be given in front of the class, but in the privacy of a separate room. In big schools, reformatories etc. this could be either a designated punishment room, e.g. the so-called "library" in Eton, or simply the principal's office.
Public vs. private BDSM play
BDSM play typically takes place either in privacy or a semi-public setting, for example in a spanking party or in a BDSM dungeon with other kinksters present. The presence of observers who are not involved in the scene is usually considered to make a punishment "public". For many spankees, public punishments have an increased humiliation factor. This may make them more or less attractive, depending on the attitude of the spankee towards humiliation.
The presence of an audience can also disturb the scene, for example if they make inappropriate noises or remarks that interfere with the player's headspace. Most play areas in dungeons are made to facilitate a limited audience watching, but sometimes they have separate rooms that allow for privacy too.
Public vs. private in spanking art
When it comes to depictions of spanking in art, literature, and videos, both publicity and privacy can have their special attractiveness.