How do I know if my EEG signal need denoising?
It primarily relies on,


what are you looking for in the EEG: define the frequency band that
relates with the phenomena you are interested to investigate and
filter out the rest (e.g. for sleep events bandpass ~= 0.8-30Hz).
power line of your area: notch filter at 50Hz or 60Hz.
Study design: if you are investigating events ensemble averages will remove random noise processes and highlight potential EEG events of interest.


Moreover, you can apply baseline removal to de-noise your signal from saturation trends and other systemical artefacts. Other experimental methods include ICA, which you will not be able to apply robustly since you only have two channels.