According to the definition of sys.transmission_queue you need to be member of the db_owner role.

Check out the WHERE clause below:

sp_helptext 'sys.transmission_queue'

CREATE VIEW sys.transmission_queue AS

SELECT conversation_handle = S.handle,

to_service_name = Q.tosvc,

to_broker_instance = Q.tobrkrinst,

from_service_name = Q.fromsvc,

service_contract_name = Q.svccontr,

enqueue_time = Q.enqtime,

message_sequence_number = Q.msgseqnum,

message_type_name = Q.msgtype,

is_conversation_error = sysconv(bit, Q.status & 2),

is_end_of_dialog = sysconv(bit, Q.status & 4),

message_body = Q.msgbody,

transmission_status = GET_TRANSMISSION_STATUS (S.handle)

FROM sys.sysxmitqueue Q

INNER JOIN sys.sysdesend S WITH (NOLOCK) on Q.dlgid = S.diagid AND Q.finitiator = S.initiator

WHERE is_member('db_owner') = 1