King Kong, the character, is not public domain.
There were two lawsuits that determined this. The second one is more important, but the first is worth noting because the judge ruled that the estate of Merian C. Cooper (Kong's creator, who directed the film for R.K.O. in 1933) had retained all rights to the character except for the two R.K.O. films and the 1932 novelization, which entered public domain when the copyright was not renewed in 1960. Because the novelization belonged the public, this would have allowed Universal to make a film based on the novel (and not using any elements exclusive to the R.K.O. films), which they ultimately didn't do. Cooper would in turn sell all but publishing/print media rights to Universal.
The second case occurred in 1982. Universal sued Nintendo over "Donkey Kong," which they claimed infringed on their rights to the Kong character and its name (which is a trademark, not a copyright). This is from where the confusion in this thread stems.
The court ruled that there was no infringement, in part because the characters could not be confused, but also because it deemed that Universal did not exclusively control the rights to King Kong. The Wikipedia article breaks it down like this:
- RKO owned the rights to the original film and its sequel. (Note: These films are now owned by Warner Bros. via Turner Entertainment.)
- The Dino De Laurentiis company (DDL) owned the rights to the 1976 remake. (Note: U.S. distribution rights to Dino's 1976 movie are held by Paramount.)
- Richard Cooper owned worldwide book and periodical publishing rights.
- "Universal thus owns only those rights in the King Kong name and character that RKO, Cooper, or DDL do not own."
- The 1932 novelization is still and will always be public domain.
The 1982 ruling didn't mean that Universal no longer had any rights to the character, just that the infringement they claimed Nintendo had committed did not apply as infringement of rights Universal owned.
In summary, Universal holds the character rights. They still hold all the cards when it comes to licensing the character for future films, unless one were to base a film solely
on the 1932 novelization. Universal likewise doesn't
hold the rights to either 1933 film, either DDL film, or the publishing rights controlled by the Cooper estate, which is why books like Anthony Browne's illustrated version of the story can be published without Universal seeing a dime.
I believe the second case also determined that Universal does not
have a trademark for the name "King Kong," for various reasons. In fact, it seems that no one has a trademark on this name. This is partly why Nintendo was in the clear to use "Kong" in their Donkey Kong series and characters. I assume it would be in anyone's rights to make a character, named King Kong, that was entirely different from the giant ape for which Universal controls character rights.
Edit: Clarification on the rights of the Toho Kong films, for those interested:
Hollywood producer John Beck paid R.K.O. for the rights to the Kong character. At that time, Cooper was not recognized as the rightsholder; as noted, R.K.O. had produced the first two Kong films. Beck brought the project to Toho, who produced the film with Beck's cooperation. Beck got worldwide distribution rights except for Japan and east Asia (which Toho controlled the rights for). Beck was the one who produced the Americanization of the film. After his English version was completed, he sold his rights (including the film itself) to Universal (then "Universal-International"), which released the completed Beck version in 1963 and which still holds distribution rights today, except for those territories in which Toho does.
The Japanese version of King Kong Escapes
opens with a disclaimer that King Kong was used with permission of R.K.O. (again, at the time recognized as the character's rightsholder). The film was a co-production with Rankin/Bass, which had just made the cartoon series the movie was based on. I'm pretty sure R/B's deal with Toho was similar to Beck's in that Toho retained Asian distribution rights and rights in several other territories while R/B got at least U.S. rights. They produced an English version of the film. I'm not sure if this Americanization was done with Universal's cooperation or not, but the film was eventually distributed by Universal and it still holds the rights today.
It's coincidental that Kong's present day rightsholder, Universal, also has certain rights to the two Toho films.
Speaking of coincidences, or lack thereof: Because Toho's two Kong films were produced with American involvement, including that of R.K.O., I don't believe they ever had the option to make any more than the two films they did make with the character. However, obviously I haven't seen the original contracts with all involved parties, but I just don't think it was a coincidence that they only made two films using the character, with both films having been produced after the American side of the productions had licensed character rights from R.K.O. If Toho had planned to make any more Kong films they would have had to license the big guy from R.K.O. (until 1976) or Universal (since). We know for a fact this is what squashed Toho's plans for a Heisei era update.