Deep learning (DL) methods have the potential to be used for detecting COVID-19 symptoms. However, the rationale for which DL method to use and which symptoms to detect has not yet been explored. In this paper, we present the first performance study which compares various convolutional neural network (CNN) architectures for the autonomous preliminary COVID-19 detection of cough and/or breathing symptoms. We compare and analyze residual networks (ResNets), visual geometry Groups (VGGs), Alex neural networks (AlexNet), densely connected networks (DenseNet), squeeze neural networks (SqueezeNet), and COVID-19 identification ResNet (CIdeR) architectures to investigate their classification performance. We uniquely train and validate both unimodal and multimodal CNN architectures using the EPFL and Cambridge datasets. Performance comparison across all modes and datasets showed that the VGG19 and DenseNet-201 achieved the highest unimodal and multimodal classification performance. VGG19 and DensNet-201 had high F1 scores (0.94 and 0.92) for unimodal cough classification on the Cambridge dataset, compared to the next highest F1 score for ResNet (0.79), with comparable F1 scores to ResNet for the larger EPFL cough dataset. They also had consistently high accuracy, recall, and precision. For multimodal detection, VGG19 and DenseNet-201 had the highest F1 scores (0.91) compared to the other CNN structures (≤0.90), with VGG19 also having the highest accuracy and recall. Our investigation provides the foundation needed to select the appropriate deep CNN method to utilize for non-contact early COVID-19 detection.