The French visa (or a residence permit) does allow you to stay up to 90 days in another Schengen country. Therefore, if your internship is shorter than 90 days, you don't necessarily need to apply for a visa or residence permit in Germany or Sweden. However, if your internship would require you to be in another country for more than 90 days, you definitely need to secure some residence title there.
A French visa or residence permit doesn't automatically allow you to work in another country in the EU. Whether an internship count as work and requires a specific authorization depends on local rules and the details of the internship (research at a university, paid internship in the private sector, etc.). In Germany, you would typically require permission from the Bundesagentur für Arbeit (federal agency for employment). Technically this authorization is not a visa (and, in fact, you have to secure that before applying for a visa if you do need one). Your employer has to take care of that.
In any case, some paperwork is very likely required and you shouldn't assume your French visa is enough. You need to approach your prospective employer about that. If they are not familiar with the procedures, there is a lot of information (in German) on the website of the Bundesagentur für Arbeit. I am less familiar with the situation in Sweden but I wouldn't assume it's OK to do an internship there without prior permission either.