Both apps can show a reply's actual text on your lock screen unless you turn previews off first. Docognito cannot help here, since it is a desktop browser extension with no presence on a phone at all. The fix lives in your phone's own Notifications settings, set separately for iOS and Android and separately again for each app.
Why the lock screen shows the reply itself
By default, both the ChatGPT and Claude mobile apps send a standard push notification when a reply finishes generating, and that notification is built to include the first line or two of the actual reply text. It behaves the same way a messaging app preview does, which is convenient when the content does not matter and unhelpful when it does.
That preview appears on the lock screen before the phone is unlocked, which means it is visible to anyone who glances at the phone while it is sitting on a desk or lights up during a meeting. The fix is not inside either app. It is a phone level notification setting, and both major mobile operating systems expose it.
This is also why searching inside the ChatGPT or Claude app for a privacy toggle tends to come up empty. Neither app decides how much of its own notification to reveal on a locked screen, since that decision belongs to the phone's notification system once the push arrives. The app hands over a title and a body of text, and it is the operating system that chooses how much of that body to display before you have unlocked anything.
Turning previews off on an iPhone
Open Settings, then Notifications
From the phone's own Settings app, go to Notifications, then scroll to the app you want to change, ChatGPT or Claude.
Set Show Previews to Never
Inside that app's notification settings, change Show Previews from Always or When Unlocked to Never. This removes the message text from the lock screen while keeping the notification itself.
Repeat for the second app if you use both
iOS treats each app's notification settings separately, so setting this for ChatGPT does not carry over to the Claude app, and the reverse.
Turning previews off on Android
Android calls the equivalent setting Sensitive notifications, and its exact wording varies slightly by manufacturer and Android version. Open Settings, then Apps, select ChatGPT or Claude, then Notifications, and look for a Sensitive or Lock screen notifications toggle. Turning it off tells Android to show that a notification arrived without showing its content on the lock screen. As on iPhone, this is set per app, so both need the change if you use both assistants on the same phone.
Confirming the change actually took effect
The easiest way to check is the direct one. Ask the assistant a short question from another device, or from the same phone through a browser tab, then lock the screen and watch for the notification to arrive. With previews switched off, the lock screen should show only the app's name and a generic line telling you a notification is waiting, with no readable words from the reply itself. If any part of the actual reply is still visible, double check that the toggle was changed for the correct app, since it is easy to adjust the wrong one when both assistants are installed side by side.
Why this has nothing to do with a browser extension
Docognito cannot touch a phone in any way
Docognito installs into Chrome on a desktop or laptop computer. It has no Android or iOS version, no companion app, and no way to read or change a notification on a phone. If lock screen previews are the concern, the phone's own settings above are the entire fix, not an extension setting.
This is worth stating plainly because the two problems sound similar from a distance. Both are about a bystander seeing something you would rather they did not. But one happens on a laptop screen while you are using a browser tab, and the other happens on a phone lock screen while the phone is doing nothing at all. They need two completely separate fixes, on two completely separate devices.
Doing both if you use the assistants on both devices
If you use ChatGPT or Claude from a work laptop and also carry the app on your phone, the two settings are worth treating as a pair rather than picking one. Turn off lock screen previews for each app on your phone using the steps above, and separately keep a document shaped page open on the laptop for the moments someone might glance at that screen instead. Neither setting reaches the other device, so skipping one leaves that half of the picture exposed.
The setting on your phone, and the one on your laptop
A phone lighting up with a visible reply calls for a different fix: go into your phone's own Notifications settings and turn previews off for each assistant app individually, since no extension can reach that layer. If your concern is the laptop screen in front of you, that is the part Docognito is built for. Treat them as two settings on two devices, not one fix for both.