I never had this problem with my 2yr old but she would always find it difficult to poo in her nappy as a baby and the stress of it all forced her to tears on many occasions even though she never suffered from constipation. I found, when possible, that the moment she started to strain I would whip off her nappy and sit her on a little potty between my legs and this would help her to go with less effort. Right from the start she hated pooing in her nappy and we are lucky to only suffer from a pooey nappy about once a month, if that!
When she was 5mths old and capable of sitting up by herself, I started putting her on the potty alone. I got a fabulous one at Target which is like a chair and has a little screwed in table that sits across their lap and holds them in. My 2yr old has only just discovered how to escape from it after all this time.
Every morning we put her on it as she poos like clockwork. After every sleep I also put her on it and before a bath and bed. Now I find she is toilet training herself and will go and get her potty or go to the toilet door when she needs to go. She has also started pooing on the toilet more regularly which is good as my 3mth old isn't far from having her turn starting on the potty. We started our eldest on the big girl's toilet on her first birthday and she pooed with no trouble at all. Now she prefers going on the big toilet although she loves having time on the potty in the mornings.
Best of luck,
Peta, NSW, mum of 2 gorgeous girls