Local time processes parameterized by a circle, defined by the occupation density up to time $T$ of Brownian motion with constant drift on the circle, are studied for various random times $T$. While such processes are typically non-Markovian, their Laplace functionals are expressed by series formulae related to similar formulae for the Markovian local time processes subject to the Ray-Knight theorems for BM on the line, and for squares of Bessel processes and their bridges. For $T$ the time that BM on the circle first returns to its starting point after a complete loop around the circle, the local time process is cyclically stationary, with same two-dimensional distributions, but not the same three-dimensional distributions, as the sum of squares of two i.i.d. cyclically stationary Gaussian processes. This local time process is the infinitely divisible sum of a Poisson point process of local time processes derived from Brownian excursions. The corresponding intensity measure on path space, and similar L\'evy measures derived from squares of Bessel processes, are described in terms of a 4-dimensional Bessel bridge by Williams' decomposition of It\^o's law of Brownian excursions.