First of all, is OP sure about 'the 13th century', since the treaty between Novgorod and Norway in 1326 officially settles the 'common tributary zone' (Hansen 2005: 369-72)?

As I summarized just last week in the additional question thread of What religion did the Kven people practice in the 1300s?, Novgorod and Norway had discussed the recently arising problem in the arctic Fenno-Scandia even in the middle of the 13th century, as stated in the Saga of Håkon Håkonsson:

'This winter when King Håkon sat in Trondheim came from the east out of Russia the messengers of Alexander, King of Novgorod; His name was Michael, and he was the knight, who was at their head. They made a stir about those things as to which the stewards of King Håkon north on the Marches [Finnmark] and the Kirjalar from the wast who paid tribute to King of Novgorod had had quarrels. For they often came to blows in strife with robberies and man-slayings. Then meeting were fixed, and counsel taken how that should be stopped.......King Håkon took this counsel, that he sent men in the spring out of Trondheim and they fared east to Novgorod with the messengers of King Alexander......They fared to Bergen and so by the east way. They came that summer to Novgorod.......And the king [Alexander] received them well, and then they made peace at once between their tributary lands, so that neither should raise strife against other Kirjalar nor Finns, but that peace was not kept very long after' (HsH, Kap. 319, í: ÍF XXXII: 154f.; English translation is taken from: Dasent trans. 1894: 277f., with modifications of the spelling of some proper names).

The treaty seemed not to last much more than the decade (when the saga was written [in 1260s]), and it is not so easy to interpret the background of this agreement. Nevertheless, however, the following points are in my understanding:

• This mutual non-interference to the agents of the other party might leave room to the activity of the (half-) third (and so on) forces on the field.
• Namely, the key for Novgorod to implement the agreement was how to control de facto actions of the Kirjals, their middlemen (in theory) - in fact, both parties had difficulty in controlling the Kirjals, or the Karelian peoples who got more and more active north of the arctic circle in the late 13th and early 14th century.
• Both Norway and Novgorod were perhaps aware that neither of the parties in question did not control northern Fenno-Scandia. Otherwise, why their joint diplomatic envoys (see above) preferred the southern route under the Swedish rule (via Bergen- south to Trondheim) to the northern border in question?

The circumstances of the late 13th/ beginning of the 14th century, based on the following cited sources, also confirm these points. Both Norway and Novgorod indeed had difficulty in collecting the tribute, but not necessarily hindered by the other's direct agent:

• 'Then, the Karelians (Kereliars) and the Kvens (Kvenir) got raid much in Hålogaland' (IA IV, a. 1271, in: Storm utg. 1888: 138): Was the Kvens not included in the Kirjals-Karelians? Anyway, this is the last allusion to them in medieval West Norse sources.

• 'The Karelians took Þorbjörn skéni, the local official (sysselman) of King Magnus [in custody], and in the mountain area they also killed 35 of this Þorbjörn's menn in front of him' (IA IV, a. 1279, in: Storm utg. 1888: 139)
• 'King Håkon [V] sent Gizurr galli to Finnmark to collect the tribute/ the tax that had not been collected for many years.....Gizurr galli left Finnmark with the tribute to King Håkon' (IA IX, a. 1310, in: Storm utg. 1888: 392)
• 'The Korel people [the Karelians] killed the townsmen in the Korel town who were Russian and brought in Nemtsy to themselves. And the men of Novgorod with the lieutenant Fedor went against them, and the Korel people surrendered, and the men of Novgorod killed the Nemtsy and the Korel traitors' (Beazley & Shakhmatov trans. 1914: 119) - The Karelians sometimes revolted against the Novgorodians even in the 14th century.

In other words, Novgorod apparently could not control every action of their alleged agent, the Karelians on the field at least at least by 1326, and the 4th middlemen, Swedish Birkarls (see my recent post) would also come into scenes. Returning to Op's original question, however, neither party seemed to negotiate the total amount of tribute with another party under such a situation.

References:

• Beazley, C. R. & A. A. Shakhmatov (trans.), The Chronicle of Novgorod 1016–1471. London, 1914.
• Storm, Gustav (utg.). Islandske annaler indtil 1578. Christiania, 1888.
• Sverrir Jakobsson, Þorleifur Hauksson & Tor Ulset (utg.). Hákonar saga Hákonarsonar, 2 bd. Reykjavík: Hið íslenzka Fornritafélag, 2013. ÍF XXXI-XXXII. [Dated Eng. trans. G. W. Dasent(trans.), The Saga of Hacon, London, 1894]

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• Hansen, Lars I. 'Fra Nöteborgsfreden til Lappekodicillen, ca. 1300-1751: Folkegrupper og statsdannelse på Nordkalotten med utgangspunkt i Finnmark'. I: Grenser og grannelag i Nordens historie, red. Steinar Imsen, ss. 352-86. Oslo: Cappelen, 2005.
• Pape, Carsten. "Rethinking the Medieval Russian-Norwegian Border." Jahrbücher Für Geschichte Osteuropas, Neue Folge, 52, no. 2 (2004): 161-87. Accessed July 25, 2021. http://www.jstor.org/stable/41052752.