If you are talking about viewing in a different window or browsers, then you need to delete msg.socketid after the switch node.
e.g
[{"id":"1abae639752d75b0","type":"inject","z":"ca26d812e65b8a9e","name":"Left","props":[{"p":"payload"}],"repeat":"","crontab":"","once":true,"onceDelay":0.1,"topic":"","payload":"true","payloadType":"bool","x":190,"y":140,"wires":[["d4ef44d286dbd4d9"]]},{"id":"d4ef44d286dbd4d9","type":"switch","z":"ca26d812e65b8a9e","name":"","property":"payload","propertyType":"msg","rules":[{"t":"true"},{"t":"false"}],"checkall":"true","repair":false,"outputs":2,"x":430,"y":200,"wires":[["3d8646aeac535a52"],["a2a1b23a93567f3a"]]},{"id":"4dfc1f3f16423c90","type":"inject","z":"ca26d812e65b8a9e","name":"Right","props":[{"p":"payload"}],"repeat":"","crontab":"","once":false,"onceDelay":0.1,"topic":"","payload":"false","payloadType":"bool","x":190,"y":260,"wires":[["d4ef44d286dbd4d9"]]},{"id":"e6c9b8eeb7086196","type":"change","z":"ca26d812e65b8a9e","name":"","rules":[{"t":"delete","p":"socketid","pt":"msg"}],"action":"","property":"","from":"","to":"","reg":false,"x":260,"y":200,"wires":[["d4ef44d286dbd4d9","8eee26c79ea65857"]]},{"id":"3d8646aeac535a52","type":"template","z":"ca26d812e65b8a9e","name":"Left","field":"style","fieldType":"msg","format":"html","syntax":"mustache","template":"img {\ndisplay: block;\nmargin-left: 1px;\nmargin-right: auto;\nmargin-top: auto;\nmargin-bottom: auto;\n}\n","output":"str","x":550,"y":180,"wires":[["3d9c243525f1564b"]]},{"id":"a2a1b23a93567f3a","type":"template","z":"ca26d812e65b8a9e","name":"Right","field":"style","fieldType":"msg","format":"html","syntax":"mustache","template":"img {\ndisplay: block;\nmargin-left: auto;\nmargin-right: 1px;\nmargin-top: auto;\nmargin-bottom: auto;\n}","output":"str","x":550,"y":220,"wires":[["3d9c243525f1564b"]]},{"id":"fd676e5bd54ea90a","type":"ui_switch","z":"ca26d812e65b8a9e","name":"","label":"Cambio Palla","tooltip":"","group":"127399c0a25c2697","order":1,"width":0,"height":0,"passthru":true,"decouple":"false","topic":"topic","topicType":"msg","style":"","onvalue":"false","onvalueType":"bool","onicon":"","oncolor":"","offvalue":"true","offvalueType":"bool","officon":"","offcolor":"","animate":false,"className":"","x":70,"y":200,"wires":[["e6c9b8eeb7086196"]]},{"id":"8eee26c79ea65857","type":"debug","z":"ca26d812e65b8a9e","name":"debug 2488","active":true,"tosidebar":true,"console":false,"tostatus":false,"complete":"true","targetType":"full","statusVal":"","statusType":"auto","x":530,"y":100,"wires":[]},{"id":"3d9c243525f1564b","type":"ui_template","z":"ca26d812e65b8a9e","group":"fec001471864780b","name":"Divisore","order":5,"width":"5","height":"2","format":"<style>\n {{ msg.style }}\n</style>\n\n<img src=\"https://cdn-icons-png.flaticon.com/128/8622/8622554.png\" style=\"width:25%;\">","storeOutMessages":false,"fwdInMessages":false,"resendOnRefresh":true,"templateScope":"local","className":"","x":700,"y":200,"wires":[[]]},{"id":"127399c0a25c2697","type":"ui_group","name":"Impostazioni","tab":"06b25a1845bda19b","order":1,"disp":true,"width":"6","collapse":false,"className":""},{"id":"fec001471864780b","type":"ui_group","name":"Display","tab":"035e2089087f4c3d","order":1,"disp":true,"width":"6","collapse":false,"className":""},{"id":"06b25a1845bda19b","type":"ui_tab","name":"Impostazioni","icon":"dashboard","disabled":false,"hidden":false},{"id":"035e2089087f4c3d","type":"ui_tab","name":"Display","icon":"dashboard","disabled":false,"hidden":false}]
Also deselect Add output messages to stored state
as this may cause issues .