Unless you are using ShipStation's Checkout Rates feature (available for Shopify, Magento, Wix, and BigCommerce), ShipStation does not currently send rates to your selling channel for your customers to select on checkout.
Many selling channels offer plug-ins you can connect to your carrier accounts to that will display live rates to your customers at checkout. If you connect the same carriers to your selling channels as you connect to ShipStation, the live rates provided should be very similar.