Answer: This has been a bad year for tomatoes, at least here in New England! All sorts of diseases took hold in my garden, with all the wet weather we had early in the season.
The spots you describe could be a number of things. If they occur on the bottom of the peppers and tomatoes, it could be blossom end rot. This is caused by fluctuations in water availability--a drought followed by a heavy rain, for example. In this case, the problem won't come back next year unless those conditions arise again.
If it is a disease problem--and there are many possible diseases--then yes, it could overwinter. Diseases occur naturally in the soil or their spores are blown in on the wind. To be on the safe side, I would clean up the garden very well at the end of the season, removing all plant material including rotting fruits. Destroy the debris rather than composting it. And be sure to rotate your crops. Tomatoes and peppers are in the same family (along with eggplant and potatoes) so don't plant any of these crops in the same bed more than once every three years.
Better luck next year!
Q&A Library Searching Tips