I had my left side removed and I was on no medication. The other 'good' side normally will take up the slack of the removed side. It usually takes quite a few weeks for any levels you had pre-surgery to go down. The good side then takes up the slack and often it goes through an adjustment period to get it right. So you will more than likely go hypo after about 4 weeks. This does not mean you will stay hypo.
It takes around 6 to 8 weeks before it it obvious what your 'good' side is doing, and you will have to have the test, laura above mentioned to establish that.
It really depends on the outcome of the side taken out and your thyroid condition in the first place as to when or if they remove the other side. Each patient is different and no two diseases are the same outcome. I had my other side removed 5 months later for a precancer, Hashimoto's and multinodular goitre!
I went hypo after the first side was taken out and remained so till the 2nd surgery. But as I said that is not always the case. I had Hashimoto's to start with!
How long does it take for symptoms to appear if you are going to have hypothyroidism? And what are the symptoms?
You have a good chance at becoming hypo with a partial removal. It is very very important to make sure your doctor keeps a very close eye on your levels for a period of time after your surgery. He might even start you on low dosage.
The most important tests to have run will be a TSH, Free T4 and Free T3.