how to check if a procedure is running, and what permissions are required?
I create my PROCEDURE_TO_TEST in a very simple way for testing purposes:

    USE MY_DATABASE
    GO

    CREATE PROCEDURE DBO.PROCEDURE_TO_TEST 
        AS BEGIN

        SELECT 'BEGIN'
        WAITFOR DELAY '00:00:15'
        SELECT 'AFTER'

        END
GO


I execute the stored procedure PROCEDURE_TO_TEST and on a separate window I
use the script below to check if it is running. (if it shows up, then it is running)

declare @object int
SELECT @object=OBJECT_ID('DBO.PROCEDURE_TO_TEST')
SELECT @OBJECT

    Select *
                from        (

                                SELECT *
                                FROM sys.dm_exec_requests
                                where sql_handle is not null

                ) r 
                CROSS APPLY sys.dm_exec_sql_text(r.sql_handle) t
                where t.objectid = @object


I grant the required permissions to MY_USER:

use master
go
GRANT VIEW SERVER STATE TO MY_LOGIN 
GO
use MY_DATABASE
go
GRANT VIEW DEFINITION ON [PROCEDURE_TO_TEST] TO MY_LOGIN 
GO


Because I use OBJECT_ID (Transact-SQL)


  A user can only view the metadata of securables that the user owns or
  on which the user has been granted permission. This means that
  metadata-emitting, built-in functions such as OBJECT_ID may return
  NULL if the user does not have any permission on the object. For more
  information, see Metadata Visibility Configuration.


And sys.dm_exec_sql_text (Transact-SQL)


  Permissions Requires VIEW SERVER STATE permission on the server.


Now for testing purposes I will run the procedure PROCEDURE_TO_TEST and on a different window I will check if it is running, IMPERSONATING MY_LOGIN.

to do this I use:

EXECUTE AS LOGIN='MY_LOGIN'


AND 

--===================================================================================
-- check who is the user running the show, and the original login
-- this is to test the 'execute as login' below
--===================================================================================

DECLARE @User VARCHAR(20)
SELECT @USER = SUBSTRING(SUSER_SNAME(), 
                         CHARINDEX('\', SUSER_SNAME()) + 1, LEN(SUSER_SNAME()))
SELECT   @USER 
        , SUSER_SNAME()
        ,SYSTEM_USER
        , USER_NAME()
        , CURRENT_USER
        , ORIGINAL_LOGIN()
        , USER
        ,SESSION_USER


And when I have done the tests, and found out the user can run the script then I come back to be me:

REVERT


now I wrap the script into a procedure that I create called isSPRunning that receives as parameter @ProcedureName and check if @ProcedureName is running:

USE MY_DATABASE
GO

GO
CREATE procedure issprunning 
@ProcedureName VARCHAR(108)
AS
/*
=======================================================================
Script  :   procedure issprunning 

Desc    :   

  Take as parameter a procedure name and returns 1 if it is running
   and 0 if it is not running or the procedure has not been found
   or the user does not have required permissions.


Required Permissions:

   The required permissions to make use of this procedure are:

   1) procedure to run this procedure
   2) VIEW SERVER state
   3) view definition on the @ProcedureName procedure

   the granting of permissions:
   (please replace the username and database name accordingly)


    use master
    go
    GRANT VIEW SERVER STATE TO MY_LOGIN 
    GO
    use MY_DATABASE
    go
    GRANT VIEW DEFINITION ON [PROCEDURE_TO_TEST] TO MY_LOGIN 
    GO
    GRANT EXECUTE ON isSpRunning TO [MY_LOGIN] 
    GO

Usage   :  

         DECLARE @I INT
         EXEC @I = issprunning 'PROCEDURE_TO_TEST'
         SELECT @I


Links   :

=======================================================================
History
Date             Action        User                Desc
-----------------------------------------------------------------------
03-Mar-2017     created       Marcelo Miorelli


=======================================================================
*/
BEGIN 
SET NOCOUNT ON

declare @object int
SELECT @object=OBJECT_ID(@ProcedureName)
--SELECT @OBJECT

RETURN CASE WHEN 
EXISTS(

                Select *
                from        (

                                SELECT *
                                FROM sys.dm_exec_requests
                                where sql_handle is not null

                ) r 
                CROSS APPLY sys.dm_exec_sql_text(r.sql_handle) t
                where t.objectid = @object

) THEN 1 ELSE 0 END 

END 
GO
GRANT EXECUTE ON dbo.issprunning to [MY_LOGIN] as dbo
GO


In order to make it easier, so that I can pass any procedure name that exists on MY_DATABASE I have extended a the permissions granted, but consider the implications of this in your own environment.

    use MY_DATABASE
    go
    GRANT VIEW DEFINITION TO [MY_LOGIN] 
    GO