Okay, I know from experience how frustrating potty training a 3 year old little boy can be. Our doctor told us at his 3 year check up that if he wasn't potty trained yet, we could assume that he was resistant to the idea. I just wanted to laugh at the doctor. I really didn't need him to tell me that.
Honestly, it sounds like he is not quite ready (emotionally) to do it on his own all the time. He will get there. I, also, have to say that at this phase in potty training, for us at least, pull-ups were a godsend. He could pull them up and down but I didn't have to clean up the puddles all over the house or wash extra loads of laundry because of all the accidents. Others may tell you that it will prolong process or set him back because he will use it like a diaper.
Here is my advice. For your peace of mind (especially with a 1 year old), put him in the pull-ups and continue to remind him. Also, come up with a reward system. Hershey's kisses worked wonders for us. For each successful attempt on the potty that resulted in pee he received 1 kiss, for poo he received 3 kisses. It was the only time he was able to have the kisses.
But here is the kicker. Nothing we did seemed to help. Our guy still seemed to refuse to actually go on the potty. He would sit there for long periods of time and try but not actually go. All of a sudden, one evening he came running into the kitchen (in November) and told me that he had to pee. I told him to hurry up and get to his potty. He did. Since then he rarely has an accident. When he does, it seems to coincide with when he is not feeling well. It took another week and a half before he started pooping on the potty. Again, it was like someone had turned a switch on in his brain and he finally got it. Since then he has had one accident with poop. That again was when he was sick.
So hang in there. It probably won't be easy but it will come. A friend of mine has 3 kids. Two out of 3 were potty trained by about 18 months of age. The other one resisted. Her technique for them was to start at about 1 year old and start putting them on the potty for a couple of minutes at each diaper change. They started associating what they did in their diaper with the potty and started going on their own to the potty early. Just something to think about with your 1 year old.
Good luck. Feel free to email me if you need to work out frustration over the potty training, etc.