Yes, at least when Firefox is set up to allow them in the Options/Preferences > Privacy & Security section (about:preferences#privacy).

Safe Mode is not a mode that should keep you safe in several ways, e.g. by not allowing cookies, but to troubleshoot issues mostly caused by external software (and particular settings). See the Diagnose Firefox issues using Troubleshoot Mode support article for more details.

I think you want to find out about'

Private Browsing - Use Firefox without saving history

Firefox Private Browsing is great for viewing websites without saving things like cookies, temp files, and a history of the pages you visit.

Why does the Private message state "when you browse in private window, fire does not save.. One of the items is cookies?

Note that in Private Browsing mode all data is kept in memory and not stored in the profile folder. Cookies work the same in PB mode like in regular mode, but are kept in a separate in-memory cookie jar and purged when you close all PB mode windows.

The Firefox description says no cookies are saved.

Then that message should be corrected?

mace2 said

The Firefox description says no cookies are saved. Then that message should be corrected?

What does the word "saved" mean to you?

In a private window, cookies are accepted and sent normally while you have private windows open. They are treated like session cookies instead of "persistent" cookies, and are not saved to disk.

I guess if you interpret "save" to mean "store in memory during your private session," then cookies are "saved" for some period of time. That's not what the page means.

Do you have a better way to explain it?

I understand what you are describing jscher2000.

Can you give an example of a sit that saves cookies to disk?

Perhaps it would be better if you explain your concern? Private browsing mode simply erases all data created during that session, when you close Private browsing, it's gone. Regular browsing keeps that data (unless you change your settings to prevent that).

Cookies also aren't all that bad, a lot of people think they are magical universal tracking devices that can do everything, but they aren't that scary.

mace2 said

Can you give an example of a sit that saves cookies to disk?

Cookies are bits of text servers ask Firefox to return with future requests to maintain continuity in your use of the server. Firefox can hold them in memory temporarily or can save them to disk. Persistent cookies set in regular sessions are supposed to last for an extended period of time (many hours or many months). Examples include cookies you allow so that you can stay logged in to a site in future browsing sessions. Firefox will save those persistent cookies to its cookie database on disk (cookies.sqlite file).