I would like to compare two competing models. When I estimate a model, I get, among others, output that says “Log data density is 1100.27”.
Is it the marginal density of the data conditional on the model multiplied by prior distribution? If not, how can I compute this product?

It is the (logged) density of the sample conditional on the model, that is the integral of the posterior kernel (the likelihood times the prior density) with respect to the estimated parameters. This is the quantity you need to compare different models.