SELECT @@VERSION 
Microsoft SQL Server 2012 (SP4-GDR) (KB4057116) - 11.0.7462.6 (X64) 
	Jan  5 2018 22:11:56 
	Copyright (c) Microsoft Corporation
	Standard Edition (64-bit) on Windows NT 6.1 <X64> (Build 7601: Service Pack 1) (Hypervisor)
 

USE myDb
ALTER PROCEDURE [dbo].[user_send_mail](@from_address varchar(64), @recipient varchar(2000), @subject nvarchar(255), @body nvarchar(max), @errnumber int OUTPUT, @errmsg nvarchar(2000) OUTPUT)
 -- WITH EXECUTE AS 'Domen\User1'
AS
BEGIN
	DECLARE @mail_id int
	SELECT @errnumber=0, @mail_id=0
	BEGIN TRY
		EXEC msdb.dbo.sp_send_dbmail 
			@profile_name = 'DbMail BOSS-1', 
			@recipients = @recipient,
			@from_address = @from_address,
			@subject = @subject,
			@body_format = 'TEXT',
			@body = @body,
			@mailitem_id = @mail_id OUTPUT
	END TRY

	BEGIN CATCH
		SELECT @errnumber=ERROR_NUMBER(), @errmsg=ERROR_MESSAGE()
	END CATCH
	RETURN @mail_id
END
go

USE msdb;
CREATE USER [Domen\User1] FOR LOGIN   [Domen\User1];
EXEC msdb.dbo.sp_addrolemember @rolename = 'DatabaseMailUserRole', @membername = [Domen\User1] ;
GRANT EXECUTE ON msdb.dbo.sp_send_dbmail to [Domen\User] ;
go

User MyDb
declare @errnumber int, @errmsg nvarchar(2000)
execute [dbo].[user_send_mail] 'no_reply@Domen.ru', 'User1@Domen.ru', N'Тестовое сообщение' , N'--Тестовое сообщение--', @errnumber output,@errmsg output
SELECT @errnumber,@errmsg

The EXECUTE permission was denied on the object 'sp_send_dbmail', database 'msdb', schema 'dbo'.

USE msdb
GRANT EXECUTE ON msdb.dbo.sp_send_dbmail to [Domen\User1] ;

EXECUTE AS login = [domain\user1]
.....
REVERT