Can multiple currencies be managed on CRM? Can each user specify their personal currency?

Yes. it is possible to manage multiple currencies on most of the CRM tools. Any user can specify any currency that they wish to use. But they are generally created by working in tandem with the base currency. By default, CRM usually ask organizations to determine a base currency at the time of creation. For any other currency, related exchange rates can be saved. 

Every user can also have a currency associated with them in the user settings. This acts as a default for all the records created by them.

Only active currencies can be used in currency amount fields. Each user has a personal currency, which is used as the default currency. Users can also create and enter amounts using other active currencies.