Should I ask for permission from the journal before making available publicly the (anonymised) text of the evaluation I received from peers? The question says it all. To add some relevant information, I must note that 


I published some articles in national journals
The level and the process of peer review in some of those national journals was quite low or even nonexistent; yet they claim to be double-blind peer-reviewed and are in the top of their field according to a journal list our Academy of Sciences use when evaluating research performance
I intend to make available publicly the substandard reviews, or, in the case of non-existing external review processes, my correspondences with the handling editor on the internet


Now it seems obvious for me that no permissions will be given to me by the journals if I ask for it, so I'm not sure whether I should ask them anyway. But would that be a bad move; legally speaking?
Unless you signed a very peculiar publication agreement, you are under no obligation to keep the reviews your received confidential.  Moreover, there is probably no issue with copyright infringement either.  Publishing a modest amount of text, for the purpose of critiquing it, almost certainly qualifies as "fair use" of potentially copyrighted material.  (This could, however, vary from jurisdiction to jurisdiction.)

So I think there is rather little legal risk of doing this.  However, it sounds to me like a very poor idea nonetheless.  In most academic contexts, the dangers of doing something wrong come less from formal legal repercussions than from the damage that may be done to your professional reputation.

Presumably everyone who has published in these journals understands the relatively lax nature of their peer review (having gone through the process themselves).  Unless one or more of the journals has an undeservedly high reputation (which seems doubtful, since you describe them as "national"; prestigious journals typically have international contributors and audiences), publishing accusations of laxity against the journal are not likely to affect the journal at all.  On the other hand, doing something like that can definitely get you a reputation as an obnoxious troublemaker, which can be extremely damaging to your career.