Abeysekera founded the Women and Media Collective in Colombo in 1984. The group promotes women's rights and has been involved with the National Women’s Charter, the National Action Plans for Women and the Migrant Rights Policy. In 2005, it helped draft the Domestic Violence Act. She became head of the INFORM Human Rights Documentation Centre in 1990, at a time when the events of the Sri Lankan Civil War were escalating. The group monitored human rights abuses on all sides of the conflict, being treated with suspicion by both the ruling government and the Liberation Tigers of Tamil Eelam (LTTE). This then resulted in death threats against Abeysekera personally and she was forced to spend some time in the Netherlands. She went back into exile between 2009 and 2010, supported by the Institute for Social Studies as part of their "Scholar At Risk". This is designed to help scholars threatened with violence in their own country.