Are the secret chat or hidden chat features in these apps actually private?

Depends on the app. Telegram Secret Chats are genuinely end to end encrypted and device-locked, that is real. But a lot of apps' hidden chat is just a PIN-protected folder on your phone, the messages are not extra encrypted, they are only hidden from someone glancing at your screen. Read what it actually claims.

Solved the same thing by switching between wifi and mobile data, oddly enough.

Been meaning to ask this for ages, so thanks.

Same here, glad someone finally asked this one.

Rule of thumb: if it says end to end encrypted, it is protecting the message in transit. If it just says hidden or locked, it is only hiding it locally. Two very different things that marketing loves to blur.