Overall, there has been inconsistent data on the effects of using zinc lozenges to treat the common cold. Most studies have also been done on children, which may lead to more uncertainty about its effects on adults, but some of the studies I will mention have been tested on adults.
One study showed the effects of using zinc to prevent the common cold in schoolchildren. They found that between the two groups they studied (one took zinc sulfate tablets, the other placebos) that the zinc-supplemented group had less of a chance of getting the cold. They concluded that zinc supplements have a beneficial effect on children.1
Another study (done by the same researchers as the second study I linked to) tested the effects of zinc lozenges on treating cold symptoms. They found that taking zinc lozenges within 24 hours of first getting cold symptoms may reduce the length of the cold. They do not give any firm recommendation though, because of insufficient data. They also mention the same side-effects as above.2 This study is also backed up by other similar study which found that zinc lozenges reduced the duration of colds by about 7 days, again with only minor side effects such as bad taste and mouth irritation.3 Other similar reports' results showed that zinc lozenges not only reduce the duration of colds, but also the severity.4, 5
Some studies say that zinc lozenges do not have any noteworthy therapeutic effect in treating the common cold.6, 7
Should you take zinc lozenges?
No studies have provided sufficient information to prove or disprove beneficial effects of taking zinc lozenges to prevent and treat the common cold. The consumption of zinc lozenges doesn't present any dangerous or deadly effects, unless you consider bad taste and nausea deadly. As of now, experts recommend talking to your doctor to decide if taking zinc
lozenges is right for you.8
What is the best type of zinc lozenge to take?
Well first, as I said above, there is no conclusive evidence. There is also no definite best type of zinc lozenge to take. Of the studies that showed positive results, zinc gluconate was used the most, but zinc sulfate and zinc acetate were also shown to be effective. Of the trials that showed no noteworthy effect, zinc gluconate was also the most tested type, but zinc acetate also showed no effect. According to WebMD, researchers are still trying to figure out which type of zinc lozenge is the best.
Risks of taking zinc supplements
Taking zinc lozenges hasn't been shown to produce any major side-effects. Sometimes, people taking them may experience bad taste in their mouth and possibly nausea. Nasal spray, as you mentioned is also not good. It may even cause loss of smell.
 Effects of Zinc Supplementation in Occurrence and Duration of Common Cold in School Aged Children during Cold Season: a Double-Blind Placebo-Controlled Trial
 Zinc for the common cold
 Reduction in duration of common colds by zinc gluconate lozenges in a double-blind study
 Duration and severity of symptoms and levels of plasma interleukin-1 receptor antagonist, soluble tumor necrosis factor receptor, and adhesion molecules in patients with common cold treated with zinc acetate.
 Efficacy of zinc against common cold viruses: an overview.
 Two randomized controlled trials of zinc gluconate lozenge therapy of experimentally induced rhinovirus colds.
 Effect of treatment with zinc gluconate or zinc acetate on experimental and natural colds.
 Will taking zinc for colds make my colds go away faster?
WebMD - Zinc for Colds: Lozenges & Nasal Sprays