Well, there's about as many ways to hear music and understand it as there are people.

Usually by the second hearing, I can tell by ear what mode a tune is in.

I started learning how to do this when I was in choir in high school, and I'm now in my 50s. So, I'd say, the more you do it, the easier it gets.

When I hear a tune or a song, my "ear" notices where the whole steps are and where the half steps are, and then figures out where "one" or "do" is. It makes that decision assuming that we are in Ionian, or major, most of the time. (Minor is the main exception).

95% of the time in folk music, the last note you sing will be the tonic, or base note of the scale ("do" or "one").   

I use Pied Piper's method of visualizing modes, but I use numbers rather than "do, re, mi" to file tunes, so I see in my mind where they start and end on a number system. That's just me.

One class I took taught us to recognize different intervals by tagging them with popular songs.
1 - 2 My coun-try (t'is of thee)
1 - 3 A song (of love is a sad song, Hi Lilli...)
1 - 4 When I (was just a lad [Lemon Tree]
    -- and so on.

(and for the purists, yes I know about Lemon Tree, but 1 - 4 is the same interval as 5 - 1 and is Lemon Tree in lydian, or does it modulate?) the trick is to train yourself to recognise the space between one note and the next. Later you can worry about modes.

Garg, I missed why Dorian runs from 'e to E, rather than 'd to D.

Blessings,
Barbara