How proficient are you at english? (native speakers welcome!)

[merithyn]

Ending like latrine? The I sound is the one I am unsure of here.

No. Latrine is more like la-treen. Doctrine is doc-trin (short i, not long e).

I've heard from both.
Is it not a case like schedule?

I've never heard doctrine said like latrine. Not to say it's wrong, I've just never heard it.

[Grey Fox]

At 1st glance, that's how I would pronounce both.

[Sheilbh]

I've heard from both.
Is it not a case like schedule?

Maybe (I'm a shed-ule person not sked-ule).

I'm aware there are words that I've picked up pronunciations of that are apparently American not English and I have no idea how or why. Process with a short o instead of a long o, I think is an example.

[Tonitrus]

It often seems that my Brit colleague, with whom I work the closest with (he is from Essex), and I, don't pronounce anything the same. 

The example that comes most to mind is the word "guard".  I say the usual American way with the "r" stressed like a pirate "g-arr-d".  For him, it will be ridiculously stressed on the "a", as in "g-AWD-rd")...with the "r" hardly heard at all.  Almost like "god" but stressing "o" to an extreme level...and a bit of a flourish.