The Kaczmarz and Gauss--Seidel methods aim to solve an linear system by iteratively refining the solution estimate; the former uses random rows of to update given the corresponding equations and the latter uses random columns of to update corresponding coordinates in . Recent work analyzed these algorithms in a parallel comparison for the overcomplete and undercomplete systems, showing convergence to the ordinary least squares (OLS) solution and the minimum Euclidean norm solution, respectively. This paper considers the natural follow-up to the OLS problem---ridge or Tikhonov regularized regression. By viewing them as variants of randomized coordinate descent, we present variants of the randomized Kaczmarz (RK) and randomized Gauss--Siedel (RGS) for solving this system and derive their convergence rates. We prove that a recent proposal, which can be interpreted as randomizing over both rows and columns, is strictly suboptimal---instead, one should always work with randomly selected columns (RGS) when (\#rows \#cols) and with randomly selected rows (RK) when (\#cols \#rows).