Take a look at how other companies with similar problem resolved it, for example: AirBnB, Uber, TaskRabbit and others.
Every one of them, unless they were extremely lucky with timing, used a narrow geographic focus, approaching established communities and fulfilling demand for one side by hand for the first little while.
In your case you can approach established organisations like universities, churches, foundations, volunteer organisations and the like to get people that would do things and do exactly the same for the other side. Initially you need to do it by hand until it starts going viral.
Spreading the word through social media is great but it needs to be very targeted and approach people that can already find something in the application.
If there are other organizations that do similar postings, for example if there are requests for volunteering on Craigslist, you can approach them to have it posted in your application.
Narrowing on one geographic location is always easier in the beginning, because you can learn the area and all the players involved in the scene, and then maximise the output, expanding to new geographic locations when needed. It doesn't mean limiting your app, but just focusing your marketing efforts in one narrow location that can bring biggest output.
Everything that I mentioned above can be applied to offline and online social marketing. Unfortunately there is no magic solution to your problem except for bootstrapping it in the beginning and finding communities that you can leverage with biggest social impact return.