The firing started around 9.30 a.m. in Jaintia Hills district's Muktapur village followed by subsequent firing at Naljuri and Jaliakhola around 11 a.m., an intelligence official told IANS.
He said the firing lasted for several hours.
A teacher, Sari Nonglamin, was hit by a bullet in her right leg at Amdoh between Naljuri and Muktapur villages, he said.
BSF officials could not be reached for comment.
The BDR indulged in fresh 'unprovoked firing' to scare away Indian villagers from cultivating certain patches of land in these areas claiming them to be part of Bangladesh.
'The BDR resorted to firing to scare us from cultivating in our own land,' Manoj Manar, Muktapur village chief, told IANS.
Muktapur, Jaliakhola and Naljuri are about 20 km east of Pyrdiwah village, which were occupied by the BDR in 2001 for days, claiming it to be part of Bangladesh, before they were forced to retreat.
There have been several incidents of exchange of firing between the BSF and the BDR this year due to claims and counter-claims over land by both countries.
Meanwhile, panic-stricken border villagers in Meghalaya have shifted to safer grounds following Tuesday's gunfire.
'The situation is tense. Most of the people, especially women and children, have moved to safer grounds after the gunfire,' Manar said.
On June 4, Home Secretary G.K. Pillai said the joint boundary working group of India and Bangladesh would meet either in July or August to resolve all the boundary disputes between the neighbouring countries.
Of the 4,098-km-long border shared between India and Bangladesh, Meghalaya shares a 443-km border with Bangladesh, part of which is porous, hilly and unfenced and prone to frequent infiltration.
Bangladesh Prime Minister Sheikh Hasina during her visit to India had agreed to maintain peace and status quo on the border.
At present there are 551.8 acres of Bangladesh land under adverse possession of India, while 226.81 acres of Indian land is under adverse possession of Bangladesh.
The areas under adverse possession were created when East Pakistan and India demarcated the international boundary in the mid-1960s. There are 11 such areas in Meghalaya.
While Bangladesh is citing documents of 1937, the Indian side relies on land records of 1914 to support its claims.