Just a simple answer, no.

As far as I know (own experience)

Can I setup an email server and send a lot of emails from a droplet ?
Answer: YES

Do I need ask some one in Digital Ocean to send emails (forwarded emails) from droplets ?
Answer: YES

Only one remark:
Sending mails (a.k.a. creating connections to port 25) will be probably firewalled in your Droplet.
With a simple ticket, they can open it.
They will warn you about Anti-SPAM rules and other caution

Roger

@Roger I run my private mail server here in DO and I didn't asked anyone for permission. The ports where all open and I had to close them with a firewall. You can open a ticket from your Control Panel to verify that.

That is true, the email server worked fine after I opened a ticket

Just to clarify, in most cases you will not have to ask for any permission to send email before hand. There are certain instances where SMTP is blocked for new accounts in order to prevent spam. If you fall into that category, simply open a ticket with the support team and they'll get you set up.