Do 'popping' or 'clicking' sounds from the dog's joint indicate surgery is needed?

No.  These sounds often arise in the months of recovery whether or not surgery is part of treatment, and then the sounds usually fade away over time.  While these noises can be disturbing to the dog's people, they are usually not really a cause for alarm.

There are several possible causes for these poppings/clickings. The cause for these sounds often cannot be diagnosed with certainty. A torn meniscus is one possibility, but there are also several other possible causes for these sounds. Even if there is a torn or damaged meniscus, that does not mean surgery is necessarily appropriate.
---- Some vets will tell a client that noises from a dog's stifle joint mean there is meniscal injury and that this means surgery is necessary.  A vet who claims to be certain that clicking/popping noises mean your dog needs surgery is at least injudicious, and possibly dishonest.
---- If you or I went to a good orthopedic specialist because we had clicking/popping in our knee but no other symptoms (such as locking-up of the joint), this would not be seen as justification for surgery.
----- Here below is a clip from the 'Johns Hopkins' website. 'Johns Hopkins' is highly respected among medical schools in the United States.  

Patient Guide to Joint Cracking and Popping

Edward G. McFarland, M.D.
Andrew Cosgarea, M.D.
Brian J. Krabak, M.D.

Is cracking and popping of joints normal?
Cracking and popping of joints is usually normal and most of the time is nothing to be concerned about. Strangely enough the exact reason joints pop and snap is not totally understood....
These noises are increased frequently after surgery on a joint, although the exact reason is not clear.

Do these sounds need to be treated?

Whatever the cause, these sounds do not need to be specifically treated. There are no long term sequelae of these noises, and they do not lead to future problems.

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"...These noises are increased frequently after surgery on a joint, although the exact reason is not clear...."

---- Since a surgical procedure is a form of trauma, it is reasonable to conclude that other traumas to the joint could also lead to an increase in these joint noises. So it is not surprising to find people and dogs with ligament injuries developing knee/stifle clickings and poppings for unknown reasons other than having meniscal damage.

Interestingly, many dogs seem to develop these noises not at the very beginning of recovery just after the injury/surgery, but weeks or months later during their recovery. The clickings/poppings could be a normal part of the recovery process rather than a new injury.  Possibly a result of looseness in the joint after swelling reduces but before scar tissue development tightens the joint to a more near-normal condition.

Swelling?
Remember that the stifle naturally enlarges as the new supporting tough fibrous scar tissue develops. This could be mistaken for swelling due to some other cause, but it is a normal part of the recovery process which nearly universally leaves dogs with the recovered stifle permanently larger.
 
What if meniscus damage is certain?
Even if an MRI were done and this showed a damaged meniscus, that does not lead to the conclusion that surgery is necessarily required. Severe damage to the meniscus would require surgery, but this would be clear from other symptoms.  A dog who is slowly improving but has noises coming from the joint doesn't need surgery just because a surgeon says so while looking at an MRI.  When the popping or clicking in a person's joint is caused by a damaged meniscus, human docs don't usually recommend surgery in the absence of symptoms in addition to the clicking / poppings sounds. 
---- This little clip is from http://www.emedx.com/
<<< Meniscus tears are typically associated with pain along the inner side or outer side of the knee. Mild to moderate swelling will accompany this pain. Clicking, catching or locking of the knee may also be present.
Many meniscus tears will respond to rest, anti-inflammatory medication and physical therapy. >>>
 
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So, if a clicking/popping sound developed in my dog's stifle(knee) after ligament injury, 
--and he was not slowly improving with properly restricted activities,
--and he was in pain from the joint. 
--and there were other symptoms such as locking up of the joint,
then I would be thinking that the noise could be an indication of a damaged meniscus causing a problem inside the joint, and that surgery could be appropriate if physical therapy (swimming) and time did not result in improvement. 
       That is, the clicking/popping combined with other symptoms and an inability to improve would point to surgery as appropriate. 
---- Not that popping/clicking = Surgery is necessary
 
Many vets are much too quick to remove meniscal tissue on the slightest pretext.  Please see the 'Meniscus' page here at this website.  Letting a vet open up the joint is not without risk.  Dogs come out of surgery with maimed joints everyday.  And many more dogs have greatly increased arthritic changes in their stifle joints in future because meniscal tissue was removed unnecessarily. 
---- It is disturbing to people to hear these popping & clicking sounds.  They will often take their dog to a vet.  If it is the wrong vet they are likely to be told that surgery is necessary.  It is important to be skeptical about surgical recommendations.  Most dogs with clicking/popping joints don't need surgery.