If you Google this, I'm sure you'll find several articles detailing it in depth. I think the answer is-- it depends
First off, for production/publishing you're going to need at least one copy of Desktop (if you don't already). I've found the capabilities of Desktop are far superior to the controls in Server, IMHO, as there are some things you simply can't do in Server's editing feature.
As for Sever vs. Online: I've seen reports that Server is ultimately less expensive over the long haul, however this doesn't take into account the overhead of server space and so on. Tableau Online offers a ready-made environment, and their 24/7 maintenance is part of what you are paying for.
Privacy is also important. If you're dealing with HIPAA data, for example, you'll need to be certain Tableau Online (a third party) meets your company's privacy guidelines. You probably won't have this issue with Server, which is hosted on-site in your company's domain.
I looked at several options, and in the end determined that Server was the superior choice for my needs. We are still largely in the design stage, have not officially rolled out company-wide dashboards, but it's just a matter of time .
If there's no sensitive data in any of your workbook, Tableau Public is ideal and it's free. All you'd need is a Tableau Desktop license to prepare and publish the workbook to Public account.
However, if there's sensitive data then your best bet would be to have a Tableau Server/Online setup and everyone who would access the report needs a license. Here are some links that could help you decide,
Let me know if you have further questions.
See, I told you! I should've refreshed
One additional option is possible if you have a core-based license for your internal Tableau Server. Then you can allow internal guest access to your workbooks. Only licensed users will be able to save custom views. Some of our most popular reports are accessed by people who don't know what Tableau is, they just have a URL, no login required.