Shitaye clinched first place in the women's 10,000 metres while Taher finished third in the men's 3,000m steeplechase.
Their medals were Bahrain's second and third of the Games. Hassan Mahboob had earlier clinched a bronze in the men's 10,000m race on Wednesday.
In the women's 10,000m final, Shitaye stormed to the finish in a winning time of 32 minutes 35.78 seconds.
An athlete from Algeria claimed the silver in 33:06.41, while the UAE's Alia Mohammed clinched the bronze in 33:44.12.
Shitaye's medal presentation ceremony is scheduled to take place today.
Meanwhile, in the men's steeplechase final, Taher had to settle for third place after finishing his race in 8:44.20.
A Turkish athlete won the gold in 8:28.79, while Moroccan Hameed Azzedine came away with the silver in 8:37.46.
In other results yesterday, Bahrain's former World champion Youssef Saad Kamel finished fifth in the men's 800m final, narrowly missing out on another medal.
He clocked a mark of 1:48.27.
On the distaff, fellow-Bahraini Genzeb Shumi was also fifth in her 800m event. She registered a time of 2:11.00 in her race.
Bahrain is competing in Indonesia with an 11-member athletics squad. The remaining athletes include former world champion Maryam Yusuf Jamal, Belal Mansoor Ali, Mimi Belete Salem, Ali Khamis, Dejene Regassa and Mohammed Yusuf Salman.
This 10-day Islamic Solidarity Games, which concludes on October 1, is being held under the slogan "Unity and Harmony".
There are 13 sports that will be competed. The sports are athletics, aquatics swimming, archery, badminton, basketball, football, karate, taekwondo, tennis, volleyball, volleyball beach, weightlifting and wushu. Around 1,759 athletes and 624 officials will take part in the Games.
Participating nations are Algeria, Azerbaijan, Brunei Darussalam, Burkina Faso, Egypt, Indonesia, Guyana, Iran, Iraq, Kuwait, Libya, Maldives, Morocco, Oman, Palestine, Pakistan, Qatar, Saudi Arabia, Sudan, Syria, Tajikistan, Turkey, Turkmenistan, Uganda, the UAE and Malaysia.
The first edition of the Games was held in 2005 in Saudi Arabia. The second event was cancelled after it had been originally scheduled to take place in Tehran in October 2009. Azerbaijan will host the 4th Islamic Solidarity Games in 2017.
