It seems the only way to always see notifications in Activities overview with GNOME 3.10 is by writing an extension or patching GNOME Shell.
There is a bug registered in GNOME bugzilla about this problem "Message Tray isn't easily discoverable (isn't shown in Activities Overview)", see:
Also I've found another old bugreport, related to this problem:
Also I've found one extension which helps access messaging tray in GNOME Shell:
Currently only high mouse pressure activates notification area/messaging tray. Low pressure does nothing. Pressure is defined by the number of pixels past the edge of the screen. If pressure is set to 250px, you need to move your mouse 250px past the edge to activate it. This setting can be tweaked in /usr/share/gnome-shell/js/ui/layout.js:
// The message tray takes this much pressure
// in the pressure barrier at once to release it.
const MESSAGE_TRAY_PRESSURE_THRESHOLD = 250; // pixels
const MESSAGE_TRAY_PRESSURE_TIMEOUT = 1000; // ms
For example, to have the message tray appear instantaneously set MESSAGE_TRAY_PRESSURE_THRESHOLD to 0.