I am familiar with how to interpret the regression coefficients on dummy categorical variables when one chooses a reference category by dropping it. However, how does one interpret the regression coefficients on each categorical variable when all categories are included (i.e. one dummy variable for each category) and the intercept is dropped?
This depends on what other terms are in the model, but the simplest interpretation is that each dummy variable coefficient is the intercept for that group. For example if you have dummy variables for "Male" and "Female" and the other term in the model is age, then the coefficient for "Male" is the estimated value for males age 0 (or at the mean age if you center age around its mean) and the coefficient for "Female" is the estimated value for the females at age 0.