Curve25519 implements Diffie-Hellman key exchange; it doesn't sign or verify anything, it only generates a shared key via scalar multiplication. If you're looking for signatures, use Ed25519, which is an implementation of Schnorr signature algorithm on the same elliptic curve.
Here are links to some JavaScript implementations of both Curve25519 and Ed25519:
- http://www.flownet.com/ron/code/ (GPL license)
- https://github.com/tonyg/js-nacl (compiled with Emscripten)