I have transferred multiple domains with the privacy service turned on. No problem whatsoever. Every registrar worth doing business with (i.e. not GoDaddy) will forward all important mail to your real email address. Emails can take a few minutes or hours to get forwarded and/or end up in the spam folder, but that's about it.
If your registrar's privacy service is so bad that emails don't get through, transfers should be the least of your worries. Because of the new ICANN verification rules, you're at risk of losing your domain if emails don't get through to you.