Is the timezone converter really free?
Yes, and there is nothing to sign up for. No account, no email address, no trial that runs out, no limit on how many cities you add and no advertising. It is free because it is useful on its own, and because some of the people who use it will eventually want the meeting to book itself.
Where is my list of cities saved?
In this browser, on this device, and nowhere else. Nothing is uploaded to Picktime and there is no account holding it. It survives closing the tab; it does not survive clearing your browsing data, and it will not appear on another device. Copy all times gives you a copy that lasts.
Does it handle daylight saving?
Yes, and that is the main reason to use it rather than adding hours in your head. The conversion asks your browser what the offset was at that exact moment in that exact city, so a date in March and a date in July give different answers where they should, and the same answer where the city does not change its clocks.
Can I get the times out of here?
Copy all times puts the whole list on your clipboard as plain text, one line per city with the name, the local time and the short zone name. Paste it into an email, a message or a calendar invite.
Can other people see the times I picked?
Not from here. This converts times for you; it does not publish anything and nobody else can open your list. If you want people in other countries to see when you are free in their own time zone and pick a slot, that is a booking page, and it is what the free Picktime plan does.