We investigate by means of a number of different dynamical Monte Carlo simulation methods the self-assembly of equilibrium polymers in dilute, semidilute and concentrated solutions under good-solvent conditions. In our simulations, both linear chains and closed loops compete for the monomers, expanding on earlier work in which loop formation was disallowed. Our findings show that the conformational properties of the linear chains, as well as the shape of their size distribution function, are not altered by the formation of rings. Rings only seem to deplete material from the solution available to the linear chains. In agreement with scaling theory, the rings obey an algebraic size distribution, whereas the linear chains conform to a Schultz-Zimm type of distribution in dilute solution, and to an exponential distribution in semidilute and concentrated solution. A diagram presenting different states of aggregation, including monomer-, ring-, and chain-dominated regimes, is given. The relevance of our work in the context of experiment is discussed.