You will use 'se faire faire quelque chose' mostly when you are having something done by someone else.
If you just use 'se faire' it means you have done/made something yourself.
Je me suis fait/e une bonne soupe au potiron hier soir = I made myself a lovely pumpkin soup last night
Je me suis faite une belle robe = I made myself a beautiful dress
You can use 'se faire' + adjective for, to get in some expressions indicating a change -
Il se fait vieux = He is getting old
Il se fait tard = It's getting late
You will use 'se faire faire' when you have somebody to do/make something for you-
Je me suis *fait faire une belle photo de famille = I had a lovely family photo done
Il s'est *fait faire un costume sur mesure = He had a made-to-measure suit made
Elles se sont *fait construire une maison à la campagne = They have had a house built in the country
You can also use different infinitive verbs other than faire as in the wasp sting example but I would say -
Je me suis *fait piquer par une guêpe = I have been stung by a wasp
rather than your example,
'Je me fais piquer par une guêpe'
which means you have made yourself be stung by a wasp on purpose, which is a bit weird unless you are in a lab doing some tests!
In a similar vein, you would say-
Les voleurs se sont * fait arrêter par la police = The thieves were arrested by the police
No intention meant, just a passive sense- something was done to you by others.
A little grammatical note here -
*The past participle of faire is invariable when followed by an infinitive even when using the reflexive 'se faire'.
but an interesting point Liz!