Jim Feyrer and Bruce Sacerdote report that at least a few quarters out, some of the fiscal stimulus had large effects, some had zero or even negative effects, and the total effect was probably lower than expected by administration economists, but not significantly lower.
A compelling part of their results is that they find that aid to state and local government to support teachers and police apparently did not save any jobs, or in other words, had no "permanent" effect on employment. Support to low-income families, on the other hand, did. This is fairly consistent with a "permanent income hypothesis" view of the stimulus. As reported by the authors, groups with long time horizons, like state and local governments and unions, correctly perceived the relief as temporary and didn't permanently hire or save a job, preferring instead to conduct business-as-usual as they would have without the stimulus. Groups with shorter time horizons or liquidity constraints made "permanent" changes, probably by spending the cash, a different outcome than in the counterfactual with the recession but no stimulus.