Bishkek is a hub for young, open-minded Westerners with degrees in journalism, philosophy, German philology and the like. Some stay on research grants or with NGO jobs, others are working online. You will quickly get to know many of them if you go to the Couchsurfing gatherings, the Trekking Union hikes and hang out in Sierra Coffee long enough. Before long, someone will offer you a job. You can find yourself something in the following fields.
Teach at a university or language school
There are many universities in Bishkek and they are in need of teachers. The level of most of them is very low so any Westerner is considered a prestigious addition to the staff. You can teach pretty much anything, from photography to tourism. A bit of enthusiasm and knowledge is all that is required. Salary levels are low. If you speak a decent amount of English, you can always find a job teaching that.
Writing, editing & miscellaneous
The 3 main industries in Bishkek that could use your help writing and editing are NGO’s, tourism companies and universities. For the miscellaneous section, I have met people randomly arriving and finding jobs as cooks, ski instructors, programmers and journalists. Some people started their own business, from a conservation NGO or a statistics company to a hotel, coffee bar and various tourism ventures.
Permits and visas
Nobody really bothers with that since the 60-day visa-free rule. People go for a visa run once in a while to a neighbouring country, or for a winter holiday to Bali.