Cross-saving consists of transferring the save file via the internet from one device (PC/Switch) to another (PC/Switch). So loading a savegame on PC after you played on Switch, or loading it on Switch after you played on PC, is the same as loading on Switch after playing on Switch, or loading on PC after playing on PC.
All of the items you collect in a savageme belong to the character you played in that session, and will remain in that particular session's savegames. When you load a different savegame (on that other player's system or on your own), that will be a completely different game and hence no items or progress from any other game you might have played will transfer. Your profile is a collection of savegames and settings, not a collection of characters with items that you (can) play in different game sessions.