Do co-owners on an investment, filing separate taxes, add dividends to their income each?

Yes...but the problem will arise when a form 1099 for dividends is received.  That will show the total amount of dividends and probably only in the name and SSN/EIN of one person.

That person will need to file F1099(s) for the other people.

https://www.irs.gov/instructions/i1099gi#idm140041452706928

Nominee/middleman returns.

Generally, if you receive a Form 1099 for amounts that actually belong to another person, you are considered a nominee recipient. You must file a Form 1099 with the IRS (the same type of Form 1099 you received) for each of the other owners showing the amounts allocable to each. You also must furnish a Form 1099 to each of the other owners. File the new Form 1099 with Form 1096 with the Internal Revenue Service Center for your area. On each new Form 1099, list yourself as the "payer" and the other owner as the "recipient." On Form 1096, list yourself as the "Filer." A spouse is not required to file a nominee return to show amounts owned by the other spouse. The nominee, not the original payer, is responsible for filing the subsequent Forms 1099 to show the amount allocable to each owner.