Hiking on the escarpment seems to enjoys a long history of acceptance, or at least tolerance, by both countries. The exact border is not marked, and is even disputed. Hikers walking along the border, and smugglers cross right over is a different kettle of fish in my books. Official reccomendation would be to take your passport along if you are on the escarpment, but I rarely do this anymore unless I intend walking further into Lesotho from the edge. Theoretically the Lesotho Mounted Police could theoretically stop and ask you for ID, though I've personally never encountered them.
However, I had a case on a Grand Traverse once where 2 people dropped out at Sani Pass. They had their passports, but since there was no entry stamp into Lesotho in them, the border official was upset, but eventually let them through.