Read your logs. When you send the emails, do your logs show that the email is being accepted by the recipient server? if so, then the problem is on their end, but if not, your logs should show you why they're not accepting the emails.
As far as receiving emails, you need to see if the logs show that the emails are even making it to your server. If so, then the email will show up in your logs and if the incoming email doesn't show up in your logs, then only the sender logs will show why not.
If you're not the administrator of your server you should contact with your provider as it's likely we can't help.
Jeff