Multiple Sclerosis (MS) is a chronic disease affecting over 2.5 million people worldwide. Its early detection is crucial for the management and treatment of the disease. Here we present an approach for automatic MS screening based on encoded speech representations. Our methods rely on Wav2Vec2 models to extract relevant traits from speech recordings of patients, which are then fed into a Support Vector Machine. Besides employing Wav2Vec2 models pre-trained on large public corpora, we also fine-tune them on 85 hours of the target language (Hungarian) in two distinct ways: for ASR and for speaker identification. Both variations outperformed the original models and conventional methods (ComParE functionals, x-vectors, and ECAPA-TDNN). Our findings suggest that fine-tuning for the actual speaker provides more advantages than the typical approach of fine-tuning for ASR purposes. Still, we improved our best MS discrimination performance when we fused features from our two fine-tuned models.