In the book, it's not specified; we're just told (emphasis mine):
[Frodo] now saw fine lines, finer than the finest pen-strokes, running along the ring, outside and inside: lines of fire that seemed to form the letters of a flowing script. They shone piercingly bright, and yet remote, as if out of a great depth.
Fellowship of the Ring Book I Chapter 2: "The Shadow of the Past"
In the film, obviously they had to make a decision on this. From examining a number of shots of the Ring, it appears as though the first two lines were engraved on the outside, and the last two engraved on the inside.
For reference, this is the Ring inscription, as printed in the book:
In the following image (from the prologue to Fellowship), we can see the first two lines of the poem engraved on the outside:
And on this image from inside the volcano (erm...spoilers?) we can see the third line and the start of the fourth, on the inside of the Ring:
Also notable in that image is that we can see the back side of the Ring, and we don't see any writing there, where we'd expect it to be if the verse carried all the way around.