Hi,
All you're saying is normal.
The role of a custom field of the type "custom text" is to display custom text on the form.
So if it's a custom field of the table "product", it will allow you to display custom text on the product edition form (and thus that text won't appear on the product page of the frontend).
If it's a custom field of the table "item", it will allow you to display custom text on the product page of the frontend.
But if you want to be able to modify that text for different products, then what you want is a custom field of the table "product" and of the type "text" or "textarea", not "custom text".