The research direction of identifying acoustic bio-markers of respiratory diseases has received renewed interest following the onset of COVID-19 pandemic. In this paper, we design an approach to COVID-19 diagnostic using crowd-sourced multi-modal data. The data resource, consisting of acoustic signals like cough, breathing, and speech signals, along with the data of symptoms, are recorded using a web-application over a period of ten months. We investigate the use of statistical descriptors of simple time-frequency features for acoustic signals and binary features for the presence of symptoms. Unlike previous works, we primarily focus on the application of simple linear classifiers like logistic regression and support vector machines for acoustic data while decision tree models are employed on the symptoms data. We show that a multi-modal integration of acoustics and symptoms classifiers achieves an area-under-curve (AUC) of 92.40, a significant improvement over any individual modality. Several ablation experiments are also provided which highlight the acoustic and symptom dimensions that are important for the task of COVID-19 diagnostics.