What are real Sikhs’ responsibilities?

Very good question Jez Singh Ji. You used the word Real Sikh. There are two words used in Gurbani. Sikh and Gursikh. Sikh in Gurbani is:
ਸੇਵਕ ਸਿਖ ਪੂਜਣ ਸਭਿ ਆਵਹਿ ਸਭਿ ਗਾਵਹਿ ਹਰਿ ਹਰਿ ਊਤਮ ਬਾਨੀ ॥ ਗਾਵਿਆ ਸੁਣਿਆ ਤਿਨ ਕਾ ਹਰਿ ਥਾਇ ਪਾਵੈ ॥ ਜਿਨ ਸਤਿਗੁਰ ਕੀ ਆਗਿਆ ਸਤਿ ਸਤਿ ਕਰਿ ਮਾਨੀ ॥੧॥ SGGS 669
Sevak Sikh Poojan Sabh Aavehi Sabh Gavehi Har Har Ootam Baanee || Gaaviaa Suniaa Tin Kaa Har Thae Pavai Jin Satgur Kee Aagia Sat Sat Kar Manee ||1||
All the Sikhs, disciples and followers come to worship and sing the sublime word (Bani) of the God. Singing and listening is of use only to those who actually obey the teachings of Guru.
Gursikh is:
ਗੁਰ ਸਤਿਗੁਰ ਕਾ ਜੋ ਸਿਖੁ ਅਖਾਏ ਸੁ ਭਲਕੇ ਉਠਿ ਹਰਿ ਨਾਮੁ ਧਿਆਵੈ ॥ ਉਦਮੁ ਕਰੇ ਭਲਕੇ ਪਰਭਾਤੀ ਇਸਨਾਨੁ ਕਰੇ ਅੰਮ੍ਰਿਤ ਸਰਿ ਨਾਵੈ ॥ SGGS 305
Gur Satgur Kaa Jo Sikh Akhaeae S Bhalkae Uth Har Naam Dhiavai || Udham Karae Bhalkae Parabhatee Eisanaan Karae Amrit Sar Naavai ||
ਉਪਦੇਸਿ ਗੁਰੂ ਹਰਿ ਹਰਿ ਜਪੁ ਜਾਪੈ ਸਭਿ ਕਿਲਵਿਖ ਪਾਪ ਦੋਖ ਲਹਿ ਜਾਵੈ ॥ ਫਿਰਿ ਚੜੈ ਦਿਵਸੁ ਗੁਰਬਾਣੀ ਗਾਵੈ ਬਹਦਿਆ ਉਠਦਿਆ ਹਰਿ ਨਾਮੁ ਧਿਆਵੈ ॥
Upadaes Guru Har Har Jap Jaapai Sabh Kilvikh Paap Dhokh Lehi Jaavai || Fir Charrai Divas Gurabaanee Gaavai Behadia Uthadhiaa Har Naam Dhiaavai ||
ਜੋ ਸਾਸਿ ਗਿਰਾਸਿ ਧਿਆਏ ਮੇਰਾ ਹਰਿ ਹਰਿ ਸੋ ਗੁਰਸਿਖੁ ਗੁਰੂ ਮਨਿ ਭਾਵੈ ॥
Jo Saas Giraas Dhiaaeae Meraa Har Har So Gursikh Guru Mann Bhaavai ||
Guru gives definition of a GurSikh. Sikh of Guru shall remember teachings of the Guru upon rising from sleep. That could be anytime. Those who work night shift sleep during day and get up in the afternoon. Those who work day shift sleep at night and get up in the morning. Upon arising, one is to bathe to clean the body, and then cleanse the mind (Mann) in the pool of nectar of Guru’s teachings (read, listen them). Following the instructions of the Guru and living life accordingly, all evils, negativity go away. All the day one is to remember instructions of Guru. Blessed are those who remember God and instructions of the Guru with every breath and every morsel of food. To get the instructions we have to prepare ourselves by controlling our ego. Nanak begs for the company of such a Sikh (GurSikh), who remembers God, and inspires others.
It is clear now that there are two types of Sikhs. Those, who are only followers of the Guru. They come to the Guru in Gurudwara as a ritual but don’t listen and obey the instructions. Even Guru can’t help such followers. There are those who remember instructions of the Guru every moment of their life and live life accordingly. They are always in a blissful state and are called GurSikh.
For more info listen to my you tube on Sikh, Singh and Khalsa. Read my book Gurmat Doctrines Part(1) available on amazon.com for more such topics.