How to designate the position of an amine substituent in an alkyl-chain?
In general usage, the practice of omitting locants when there is no ambiguity is widespread. In particular, the locant ‘1’ is often omitted when there is no ambiguity.

Clearly, the locant ‘1’ for the amine group in unsubstituted ethanamine can be omitted since the isomer ‘ethan-2-amine’ does not exist.

Furthermore, the locant ‘1’ for the amine group can be omitted in substituted ethanamines provided that the amine group is the principal characteristic group (i.e. the characteristic group chosen for citation at the end of the name by means of a suffix or a class name), for example 2-chloroethanamine.

Hence, the omission of the locant ‘1’ for the amine group in 1-phenylethanamine is permissible in general usage. Since locants are placed immediately before that part of the name to which they relate, it is clear that that the remaining locant ‘1’ in 1-phenylethanamine refers to the phenyl group.

However, for absolute clarity in preferred IUPAC names, the current IUPAC recommendations are prescriptive about when omission of locants is permissible. According to the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book), locants are omitted in preferred IUPAC names in the cases that are mentioned in Subsection P-14.3.4, in particular:


  P-14.3.4.2 The locant ‘1’ is omitted:
  
  (a) in substituted mononuclear parent hydrides;
  
  (b) in monosubstituted homogeneous chains consisting of only two identical atoms;
  
  (…)


Thus, for example, methanamine (a) and ethanamine (b) are preferred IUPAC names.

Generally, however, no locants can be omitted if any locants are required to identify the specific isomer.


  P-14.3.3 Citation of locants
  
  In preferred IUPAC names, if any locants are essential for defining the structure of the parent structure or of a unit of structure as defined by its appropriate enclosing marks, then all locants must be cited for the parent structure or that structural unit. For example, the omission of the locant ‘1’ in 2-chloroethanol, while permissible in general usage, is not allowed in preferred IUPAC names, thus the name 2-chloroethan-1-ol is the PIN. (…)


Therefore, the omission of the locant ‘1’ for the amine group in 1-phenylethanamine, while permissible in general usage, is not allowed in preferred IUPAC names. The preferred IUPAC name is 1-phenylethan-1-amine.