The positionally caused phonetic process, particularly, reduction of a vowel at the end of a rythmic group, caused by a stress, fixed at the beginning of a rhythmic group, has affected the morphological structure of a verb in the dialects where this phonetic process occurs.
The Iterative tense/aspect forms marker is -i -i in those Georgian dialects where the Iterative forms are preserved. In the Khevsurian and Tushian dialects where -i -i at the absolute end is represented with its short variant (allophone) -i, the Aorist and Iterative II forms may coincide:

Aorist: gavakeT-i gavak'et-i (short -i) (I did)

Iterative II: gavakeT-i gavak'et-i (short -i) (I used to do)

This coincidence of the Aorist and Iterative II forms must have caused the appearance of the specific forms of Iterative II (formed by -id/-od -id/-od suffixes) in these dialects:

Aorist:
gavakeT-i gavak'et-i (short i) (I did)

Iterative II:
gavakeT-id-i gavak'et-id-i (I used to do) (Khevsurian, Tushian ), gavakeT-od-i gavak'et-od-i (I used to do) (Pshavian dialect). The phonetic process -i -i  > -i -i(short i) in the Pshavian dialect has not regular character unlike the Khevsurian and Tushian dialects where it is a regular process.

In forms of the third subject person the suffix -i -i is not at the end but is followed by the third person subject marker -s (-s). That is why -i (-i) is not reduced and, consequently, the suffix -id/-od -id/-od does not appear:
ga-a-keT-i-s ga-a-k'et-i-s (Singular)
ga-a-keT-i-an ga-a-k'et-i-an (Plural)
  (Khevsurian, Tushian, Pshavian); Sporadic examples with -od -od suffix in the third person are found (Giorgi Cocanidze, Tushuri xolmeobitis shesaxeb, On the Tushian Iterative, in: Iberiul-kavkasiuri enatmecniereba, XVII, Tbilisi, 1970).

More evidence for assuming that complicated form of iterarive (formed by the suffix -od/id) is a way for overcoming the coincidence of different tense/aspect forms, is that in the Mtiulian and Mokhevian dialects where the process -i>-i -i > -i (short i) does not occur, Aorist and Iterative II have preserved the Old Georgian forms in the first and the second persons as well as in the third person.
 

Iteratives formed by adding -id // -od suffixes in Khevsurian, Tushian and Pshavian dialects must be a phenomenon resembling (but not identical to) the process by which in Old Georgian the Iterative formed by -od suffix emerged in the 10th-11th centuries (A. Shanidze and Levan Kiknadze, Shereuli xolmeobiti dzvel kartulshi, The Mixed Iterative in Old Georgian, in: "Orioni", devoted to Akaki Shanidze, Tbilisi, 1966; A. Tchintcharauli, Khevsurulis taviseburebani, Specific Features of the Khevsur dialect, Tbilisi, Mecniereba, 1960). The -od suffix in the so-called mixed Iterative in old Georgian, unlike its lesser role in the Khevsurian, Tushian and Pshavian dialects was represented in the third person too: mi-s-c-od-i-an, instead of regular: mi-s-c-i-an (they used to give them something).

Another interpretation of these forms belongs to Arnold Chikobava who suggested that forms of Iterative forms with the -od suffix evidenced both in Old Georgian and in the modern dialects must have been derived from the archaic imperfect (A. Chikobava, Ergat'iuli k'onst'rukciis p'roblema iberiul-k'avk'asiur enebshi, Problem of Ergative construction in Ibero-Caucasian languages, II Tbilisi, 1948,). For more information see From the history of tense/aspect forms.