Not in a reasonably well built system, which will normalize for volume. However, if the actual speech production volume is different, with speakers shouting or whispering, this may have an effect as these introduce significant changes (e.g. no voicing in whispered speech). 

Hi.  I would like to add to the response by Andras.  assuming you will be using MFCCs for the features for your speaker recognition system, and assuming that you do not use the C0 and indeed do not any other computation of energy, in the 0th order, there will, in practice, still be some leakage of the volume effect into the values of the higher Cepatral coefficients.  For longer speech segments, you should be able to normalize the volume across the utterance, making sure that you do not have any clipping.  

You have to be careful though, if you do volume normalization for the whole training set, then you will always need to do normalization, both for the enrollment data and the verification or identification data.  It would be better if you did not normalize the training set and only normalized the enrollment and test data.  Beware of the short test utterances!

You can read more about in my textbook, H. Beigi, "Fundamentaks of Speaker Recognition," springer, New York, 2011.

Well, if the data are too loud it will be distorted. So first make sure there's no clipping. Also note that the source of the loudness makes a difference. Is it because the gain was too high, or were people shouting? Other things being equal the training data ought to be reasonable similar to the test data; the end this is still a pattern matching problem.

Note that techniques such as CMN (cepstral mean normalization) are useful. In our own work we haven't observed much effect of normalization: the features are spectral, so as long as that information is reasonably there, things should should work. If anything, we've noticed that attempts at normalization usually degrade performance.

Of course what you should do to find for sure in your situation; simply do different trainings and see what happens.

I don't see much difference while building the system for speech signal security.

I didn't see any difference in system performance, I tried different amplitude levels.