My lurcher, R, and I walk regularly with a Golden Retriever, S, who is Caesar Millan-ed at home. We met S at a time when R and I were recovering from the trauma of a 2-dog attack on R, leaving her bitten and me with a broken ankle.
At the time S was being CM-ed, having been 'barked at' by a greyhound, which caused her to 'dislike greyhounds'.
S’s mum and I have been resocialising our dogs carefully over a couple of months, using the method emmabeth advocates. We have now got to the stage where R and S can meet each other offlead and walk together for about an hour and a half. They meet with tails and ears relaxed. They sniff both ends and set off doing doggy things. Once we reached the stage of both dogs being comfortable offlead together, S’s mum and I have been somewhat hands-off as we were aware that any nervousness on our part (from R and I being attacked, and S being ‘barked at’ by the greyhound) would transfer to our dogs.
Having met politely, S will then shadow R during the whole walk. S uses Big Body Language and a certain amount of staring to claim ‘her’ space. R sidles past - not fearful I don't think, but deferential (I'm only little, I only want to sniff this grass, you can stand there, I'll go round the long way).
If the dogs are near when S is claiming her space, I might walk straight at S and interrupt her mind games by pushing her out of the way with my legs, but usually they are several yards away.
Although R does not seem cowed by S, she is submissive and I'd say she was being restricted by S's claiming a space such as a gateway, bridge, gap etc where she will stand like Boadacea with tail up and make R find a way around. The restricted areas (gaps) in question are not part of the route, but something like a gap in the hedge where R has gone through to check the other side. S will stand sentry at this gap, forcing R to find an alternative way back.
A couple of times now R has shown her teeth when S has adopted this You-May-Not-Pass pose. Yesterday she couldn't find her way around S, as she usually would, because there was a barbed wire fence. S wouldn't move, so R launched herself with handbag flying, causing S to leap back nonplussed. No physical harm done, just a burst of noise. S did not retaliate, she backed off and behaved herself - for a while!
S's Mum maintains that S is shadowing R to get her to play running about games. To my mind, though, S's shadowing of R is not about initiating a chase. To me it's more a case of "I'm walking beside you to let you know that you can go over there, but not over here, because over here belongs to me".
On balance I think that R gains more than she loses with these walks. When we were recovering from our attack she was very fearful of other dogs, and she has gained confidence walking with S. 6 months ago she would have bolted for home if another dog looked at her. If anything, I have taken R's 'standing up' to S as a sign that she is recovering confidence. She goes forward to greet S when they first meet rather than shrinking back. She isn't clingy on the walks, she doesn't hide behind me.
It is hurtful to me to see R continually deferring to S, but I realise that’s a very human reaction and not necessarily how R sees the situation herself.
R is 8 years old and S is 6. Both are spayed. S is walked 3 times a day and fit as far as I know. S has a reputation for being ‘nosey’ – i.e. a busybody interfering with other dogs. She is regularly put in her place by other dogs at her training class/brawl and severely CM-ed by dad, although mum is more interested in understanding her behaviour.
The question is: should we remain hands-off and let them sort out their relationship/hierarchy?
Many thanks in advance
Unfortunately the owner is a bird-brain, but you can't have everything 
