Thanks for getting back to me, @Sarah_Miller1! Please note that personal notifications are not sent when you are active in Asana. In order to test if this is what is happening, you can close all Asana tabs in your browser, restart Slack and request to your colleague to assign you a task in Asana.
If you are not active in Asana and you are still not receiving notifications, I recommend you contacting our support team to investigate this further: How to contact our Support Team ✉