After the first unlock after a reboot yes, except for those items with very high security (usually just keychain items) that require an unlock for every access. Lock screen items can be locked every time you lock the phone, or set to allow access while locked (after first unlock). Unfortunately this is an all or nothing setting, unless the widget specifically uses the redaction views to hide content.

Oddly, I find the weather widget refreshes much more often than the app. I'll tap on the widget, which is up to date, which launches the weather app, which could be from yesterday, and have to wait while it refreshes. Apparently the app doesn't implement any background refresh at all, which is really weird.