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The two agencies do talk to each other. Being on unemployment means you are able to work full time. While on unemployment you must actively be searching for employment. If you are disabled this implies you cannot accept full time employment.
If you are able to work full time you cannot be on SSDI. It is possible if one or the other agency decides you have been improperly collecting money you can be asked to pay back one or the other agency. There probably would not be a penalty if they though you made an innocent mistake, if they thought you were knowingly trying defraud social security or Unemployment there could be a significant even criminal penalty. In any case you may get charged interest.
The biggest con is that you may get charged with fraud, you will have to pay interest. If you don't have a lump sum to pay back they can put you into collection.
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