(1) Background: Diabetes mellitus (DM) is a growing challenge, both for patients and physicians, in order to control the impact on health and prevent complications. Millions of patients with diabetes require medical attention, which generates problems regarding the limited time for screening but also addressability difficulties for consultation and management. As a result, screening programs for vision-threatening complications due to DM have to be more efficient in the future in order to cope with such a great healthcare burden. Diabetic macular edema (DME) is a severe complication of DM that can be prevented if it is timely screened with the help of optical coherence tomography (OCT) devices. Newly developing state-of-the-art artificial intelligence (AI) algorithms can assist physicians in analyzing large datasets and flag potential risks. By using AI algorithms in order to process OCT images of large populations, the screening capacity and speed can be increased so that patients can be timely treated. This quick response gives the physicians a chance to intervene and prevent disability. (2) Methods: This study evaluated ConvNeXt and EfficientNet architectures in correctly identifying DME patterns on real-life OCT images for screening purposes. (3) Results: Firstly, we obtained models that differentiate between diabetic retinopathy (DR) and healthy scans with an accuracy of 0.98. Secondly, we obtained a model that can indicate the presence of edema, detachment of the subfoveolar neurosensory retina, and hyperreflective foci (HF) without using pixel level annotation. Lastly, we analyzed the extent to which the pretrained weights on natural images "understand" OCT scans. (4) Conclusions: Pretrained networks such as ConvNeXt or EfficientNet correctly identify features relevant to the differentiation between healthy retinas and DR, even though they were pretrained on natural images. Another important aspect of our research is that the differentiation between biomarkers and their localization can be obtained even without pixel-level annotation. The "three biomarkers model" is able to identify obvious subfoveal neurosensory detachments, retinal edema, and hyperreflective foci, as well as very small subfoveal detachments. In conclusion, our study points out the possible usefulness of AI-assisted diagnosis of DME for lowering healthcare costs, increasing the quality of life of patients with diabetes, and reducing the waiting time until an appropriate ophthalmological consultation and treatment can be performed.