We develop a comprehensive theory for describing the experimental beam profiles from multimode fiber Raman lasers. We take into account the presence of random linear mode coupling, Kerr beam self-cleaning and intra-cavity spatial filtering. All of these factors play a decisive role in shaping the Stokes beam, which has a predominant fundamental mode content. Although the highly multimode pump beam is strongly depleted, it remains almost insensitive to the different physical effects. As a result, the intensity of the output Stokes beam is an order of magnitude higher than the pump intensity at its maximum, in quantitative agreement with the experimental results and in contrast with the simplified balance model.