Are John the Beloved and the Three Nephites actually still on the earth? If so, what are they doing?

Yes, the Savior granted to John the Beloved and the Three Nephites their desire to tarry on the earth. What they’re doing is bringing souls unto the Lord until He comes again (see D&C 7:2; 3 Nephi 28:9).

The Savior told John that he would “prophesy before nations, kindreds, tongues and people” and become “as flaming fire and a ministering angel [and] minister for those who shall be heirs of salvation who dwell on the earth” (D&C 7:3, 6; see also John 21:20–23). Mormon said that the Three Nephites would be among the Gentiles and Jews, who would “know them not” (3 Nephi 28:27–28).

We don’t know anything more specific about the activities of these men. We know only that they have been transfigured so that they will not taste of death and that, whatever they are doing, it is for the purpose of bringing people to the Savior.