User -- a human
Master -- an odoo database where this module is installed
Client -- an odoo database where you need to authenticate user and where module auth_oauth module is installed

Steps

From client database redirects user to url1 at master database:

import urllib

url1 = '/oauth2/auth?%s' % urllib.urlencode({

'state': {"p": 1, "r": "%2Fweb%2Flogin%3F", "d": "client_database"},

'redirect_uri': 'http://odoo.example.com/auth_oauth/signin',

'response_type': 'token',

'client_id': 'f23514a1-13a2-40e5-9033-5018e7f3a052',

'debug': False,

'scope': 'userinfo'

})

print url1

if user is not logined at master:

master redirects user to /web/login page
user enters login-password
master redirects user to url1

then master redirects user back:

http://odoo.example.com/auth_oauth/signin#access_token=ksESez4jRUxg0v6Kt7HkE5Z4ZtyrrM&token_type=Bearer&state={%27p%27%3A+1%2C+%27r%27%3A+%27%252Fweb%252Flogin%253F%27%2C+%27d%27%3A+%27some-test-3%27}&expires_in=3600&scope=userinfo

then client redirects user to itself (see @fragment_to_query_string in auth_oauth module)

http://odoo.example.com/auth_oauth/signin?access_token=ksESez4jRUxg0v6Kt7HkE5Z4ZtyrrM&token_type=Bearer&state={%27p%27%3A+1%2C+%27r%27%3A+%27%252Fweb%252Flogin%253F%27%2C+%27d%27%3A+%27some-test-3%27}&expires_in=3600&scope=userinfo

then client make request to master:

/oauth2/tokeninfo?access_token=ksESez4jRUxg0v6Kt7HkE5Z4ZtyrrM

and master returns user data:

{"user_id": 3, "email": "admin@example.com", "name": "Administrator"}